At the beginning of chapter 4 of his book, King is mainly concerned with Stephen Schiffer's objection to structured propositions. He writes the objection out as follows:
(1) If any theory of structured propositions is true, then (a) 'barks' in Ralph believes that Fido barks' functions as a singular term whose referent is a constituent of the structured proposition to which the that-clause refers.
(2) If (a), then the following inference is valid:
Ralph believes that Fido barks
Therefore, (Something Exists x)(Ralph believes that Fido x)
(3) But the inference isn't even coherent, let alone valid.
(4) Therefore, No theory of structured propositions is true.
King's method of rejecting this objection is to find a way to reject (1). To do this he gives three statements which are entailed by Schiffer's argument. The successful rejection of any of these three statements, according to King, results in the defense of Structured Proposition Theory (SPT). I will give the first two claims, but no the third because I am not concerned with it at this time:
i) Structured proposition theorists, including Russellians, are committed to the claim that the referent of a that-clause is determined by the referents of the expressions in it and how they are combined syntactically (CH), and so all the expressions in a that-clause (including 'barks' in 'that Fido barks') must be referring expressions.
ii) STPs, including Russellians, are committed to the claim that that-clauses are referring expressions.
King states that in order to disprove (i) all you need to do is disprove (ii) because (i) is entailed by (ii). King's argument against (ii), as far as I can tell, is that instead of referring expresions, you can hold that a belief ascription such as 'Lucy believes that Fido barks' is true iff Lucy stands in the belief relation to the proposition that Fido barks. In this case no referring expression is needed.
Basically, King showed that there actually are that-clauses which are not referring expressions.
a) Belief ascription that-clauses ('Lucy believes that Fido barks') are not referring expressions
b) If (ii), then all that-clauses are referring expressions
c) (a), therefore ~(ii)
The reason that I focused on King's argument against (ii) is that I think there might be something fishy about it. You would think that if it were as simple as it appears above to reject (ii), Schiffer, a University level academic, would not have committed himself to such a vulnerable premise. I think the error in King's argument must lie in the nature of Belief ascription that-clauses. It does not look like they are the same type of that-clauses which Schiffer is talking about in his descriptions (even though according to Schiffer type of that-clause shouldn't matter because of how he thinks all SPT's fall prey to his objection).
I am also curious about this belief relation which King has no problem incorporating into his structured proposition theory. On page 103 King described the things, according to SPTs, that the constituents of propositions are: "objects, properties, and relations". It kind of seems like squeezing beliefs into the equation is cheating a bit. Obviously, either belief imports something into propositions which stands in the place of referring expressions (which I think King would agree with) or it is not the type of that-clause proposition Schiffer is objecting to. Other that this semi-objection to King, I think I mostly agree with his rejection of Schiffer's referrent expressionism objection.
One last note. Both Schiffer and King use variations of (2) to support their objection.
(2) Ralph believes that Fido barks, therefore (something exists x)(Ralph believes that Fido x)
Schiffer calls it incoherent and invalid. I can't figure out why this is such a problem though. Schiffer especially confuses me on the issue because of his insistence on how 'barks' is a singular term with the co-referential expression 'the property of being a barker'. So the claim above is (something exists (the property of being a barker))(Ralph believes that Fido barks ('barks' is co-referential with 'the property of being a barker' and so can be substituted in for grammarical correctness). I just don't see why this is "incoherent and invalid.