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Thanks Daniel, I just noticed your message!

> I'm still tinkering with this, so maybe I'm missing something here, but
things keep coming down to one or two questions so I'll go ahead and ask
them.

:)

> Patrik, you give the semantics as [X Y]->"in the X of Y". If matching up
brackets and applying this semantics was all there is to it, the first
example sentence, [topic [nom boy] [acc [on hill] house] see], becomes

> In the topic of see.
In the nom of boy.
In the acc of house.
In the on of hill.

> Which is the same as [topic see] [nom boy] [acc house] [on hill].

> Clearly that's not the intention. But what exactly does the nesting mean?

First to make sure, there are two languages; here you're asking about the two-category language. 

The nesting is obligatory because otherwise you wouldn't be able to know what refers to what. In this sentence everything refers to the topic - which is a seeing ‒ so other members are embedded. Likewise, it's the house that's on the hill, so [on hill] refers to [acc house] and is therefore embedded. There are two other possible places for [on hill], but each would imply a different meaning for the sentence i.e. either the boy being on the hill or the 'see' - the topic or action - being on the hill. Therefore: it's not the same.

> That's my first question. My second is, what is meant by your basic
semantic formula? The English is ambiguous. "In the 'with' of 'dog'" seems
to suggest something like "in a 'with', specifically the (only) one
belonging to 'dog'". But the example [acc house] would read "in an 'acc',
specifically the one belonging to the house". Here and elsewhere, when I
try to interpret the semantics I end up leaning toward "in an "acc"
consisting of a house" or even "in an acc., namely the house". It seems
like your preposition class isn't always asserting the existence of
something which the corresponding noun owns, possesses, has, etc. Or maybe
it is?

Like Logan said the anaphorics have to be determined wisely. If there are none, as is the case at this point, it could be anything. This is a language that lacks articles - which is quite common - so the English translation is a matter of liking. For instance, in Finnish you would use phrases like "talon vieressä" ("in the beside of the house", literally "house-of beside-in" i.e. beside the house; there's no saying it shouldn't be "in a beside of a house"). Anyway the answer should be yes indeed, for instance the house has a beside, and the dog has a with (Finnish 'kanssa' - fossilised location word + an ending that looks like the inessive). The Esperanto dictionary PIV actually has the noun 'kuno' for a 'with'. It surely is semantically possible. If you want an English place-noun, you could use company: with the dog = in the company of a dog: [company dog] = [with dog]; not to be confused with an enterprise (these could be company1 vs. company2).

Anyway, according to the definition, "the accusative of the house" means "the form of the house where the house is the object". 'Form' is probably not the best word, but it is in my English dictionary. It should be something like status or position. In Finnish we have the word 'sija' for case, an old word for place.

This is actually not the same as the house being the accusative (hence in this language you can't properly say [object house]); nor the with being the dog. The with is what's with the dog, so it can't be the same as the dog, otherwise it wouldn't make any sense. 

So, cross over: > "in an "acc" consisting of a house" or even "in an acc., namely the house".

> Giving different answers to those two questions, I've come up with what
seems like a dozen variations of the language. Many of them seem
functional, but unless I'm fooling myself they're not the same as one
another, and tend to require different definitions of the vocabulary in
order to make the example sentences work. (Or just don't work with the
example sentences.) But I figured I'd just ask rather than trying to figure
out which ones actually work.

Well, just look at the parse tree and you see it works perfectly, whether it makes any sense or not. You can express everything unambiguously with two lexical categories. That's it basically. But do keep asking!