Harry D. asks some questions. He isn't sure King can answer them. I will try to, though I may do a poor job because I don't really understand the questions in question.
(Q1) King's use of models.
(1) King says that propositions are not mathematical objects.
(2) King's motivation for (1) is Benacerrafian reasons which entail (3).
(3) King doesn't think there are any mathematical objects.
(4) But he uses diagrams for propositions.
(5) And these diagrams have to be mathematic objects. They are a certain kind of graph.
(6) But we cannot identify propositions with the graphs.
(7) So King has not answered: What are propositions?
(A1) I don't think (7) follows.
I'm not so bothered by (6). I don't think that it entails (7) I'm not bothered by introducing things via models. I might say that atoms are like little marbles that dance when heated. I don't mean to identify atoms with marbles or heating with dancing.
But maybe we still have to ask, nice model aside, what the atoms are. And so too with propositions. But if atoms and propositions are non-observables, I'm not sure how we can talk about them with the use of models and diagrams unless we give some attempt at representation. My scientific commitments are that the observables are all eliminable and reducible to the unobservable. So I'm not bothered by being able to talk about the essence of something, but not being able to pictorially show it. (6) seems to be taken to show King can only hint at what propositions are. (7) takes this to be invalid. I don't think it is invalid.
I'm not sure about (3). I don't think King denies that there are 'mathematical objects'. He certainly thinks that there are objects with which mathematics is concerned. I don't see how a structuralist / functionalist about numbers is a number-hater. I don't think that being non-platonistic about something makes you deny that it exists.
About (4): It seems like the model King uses is a representation of the proposition. A proposition seems to be a representation + semantic contents. And it seems that agents have to add the semantic contents. So we can never fully map what a proposition is in a diagram or model. But I don't see this to be a big deal.
(Q2) What is language?
(1) King never tells us what a language or a sentence of a language is ontologically.
(2) Whatever else English is, we know it is productive. One can produce novel sentences.
(3) On King's view, the propositions don't exist until the words get said.
(4) (3) should strike us as odd. The Chisholm style theory doesn't have this worry.
(5) King's account closes the door a priori on animals thinking propositionally.
(6) So King is vague about language, and his conclusions are counter-intuitive.
(A2) I think we can deny (6).
It seems easy to supply a King friendly account to attack (1). I think we can use some Sellars here to help King out. Language exists in the narrowly physical causal order. It is scribbles and squawks. But language is also in the broadly construed physical causal order. We don't just hear sounds, we hear words. We don't just understand sounds, we understand words.
I think the same problem could be posed for maps. Maps are in a sense just designs, blips, dots or scribbles. But when one knows how to read a map, the map is much richer. We don't just see the representation. We see what is represented.
About (2) - (4): Productivity doesn't seem to be an issue. Given the atomic bits of English, one can construe novel combinations. Given the rules of English, one can construe novel acceptable combinations. We don't need the propositions to have existed prior, we only needed the constituents to have. I don't need all the numbers to exist in order to do any math. I only need '1', '2', etc., '9', '0'. I can just reuse these numbers to get '12' or '124'. I don't see any problem with this.
I think (5) is a silly concern. I don't think it is bad that King closes the door a priori on animals thinking propositionally. It is still an open question, in a sense, even if we so close the door. But I think there are pre-theoretical reasons to doubt this anyways. Some people don't. There is reason, even if King is right, for one to argue that animals have an animal language or a private language, so they could think propositionally even if King were right.
(Q3) Vagueness on properties and relations.
(1) King assumes that there are properties and relations.
(2) He doesn't say what they are.
(3) He should.
(4) So it's not clear how propositions can exist in his sense, since he doesn't tell us what properties and relations are.
(5) Why cannot we think of propositions as being properties of the actual world?
(A3) I think one could deny (4), since even if he doesn't tell us what properties and relations are, it seems easy to figure out.
I guess I don't have a lot to say here. About (2): I think that the ideas of properties and relations is common enough. A property is a feature or character of a thing. A thing is hot or cold, small or large. I relation is a feature or character between things. A thing is next to another, distant from another.
One could be process ontology oriented. A 'property' like 'being tall' is a way for a thing to be intrinsically. So Wes is tall because there is a tallness going on. A 'relation' like 'being a brother' is a way for a thing to be extrinsically. So Wes is a brother because there is a thing he is related to via brotherness. These things are in space and time and enduring things so they seem non-spooky and physical.
I hope that isn't poorly stated. But it doesn't seem to be contrary to King's view.
About (5): Since propositions are representational, it seems the cannot be properties of the world in the way Harry D. wants. But maybe we can say that since they exist, like the CN Tower and my breakfast, they are properties of the world.