Okay, it's been an interesting discussion. I'm already drawing my conclusions and making changes to what I sketched for my speech.

At this point I would conclude that it's safe to say that all meaningful languages have nouns. It's possible to build all kinds of systems on top of that, but the nouns never disappear. So a language could for instance have nouns + various categories e.g. "cold" vs. "warm" words. If a language is monocategorial, it has only nouns; if it has further categories, these must be in the addition of nouns. I'm proposing this as an axiomatic truth.

Understanding is trickier. I'm doing (non-Chomskyan) minimalism, so basically these are the simplest things. But if there's anything people haven't seen before, it's because it's hard to see.

And: on nouns vs. predicates: when predicate is used as a term for a flexible class, the perspective is western/English. Your argument is not logically motivated. For simplicity, let's look at a language that has a flexible class for (or makes no formal difference between) nouns and verbs. On the surface level there are in fact three logical possibilities:

1) the language expresses them all as nouns
2) the language expresses them all as verbs
3) the language expresses them all as a class distinct from noun or verb; one that can e.g. be called "predicate".

So we can say you have a 1/3 chance to be right based on this.

Now we expand this category X to include any traditional word classes as we can do with PL notation; even English prepositions can logically take a predicate place, e.g. To(Swim,Teacher) for "The teacher goes to swim"; the structure is more like "Swim" takes the function of "To" concerning Teacher.

We can do a mini-test with this expression using hypotheses 1 to 3 above as follows:

1) All of the words can be understood as nouns. We test this hypothesis by a conversion into English nouns: Destination(Swim-event,Teacher), or: "A swim is the destination of the teacher". Sounds good, I suggest we accept hypothesis 1 as semantically plausible.

2) All of the words can be understood as verbs: "The event of swim is the event of being a destination for the event of being a teacher." Here it's not the teacher who goes swimming, but the being a teacher ‒ "teachering". It seems we can't express all meanings without nouns. I suggest we reject hypothesis 2.

3) All of the words can be understood as predicates, or members of a flexible class: The swim "to's" (functions as the "to", or the destination) for the teacher. Sounds good, I suggest we accept hypothesis 3 as semantically plausible. 

So we have two semantically plausible hypotheses: 1 (nouns) and 3 (predicates). Hypothesis 1 suggests that the structure can be expressed with nouns only. If we imagine a language with nothing but nouns, all of the words in To(Swim,Teacher) are actually nouns although from English POV they look like a preposition, a verb and a noun.

Hypothesis 3 suggests that the structure can be expressed with a flexible category which incorporates prepositions, verbs and nouns. 
We see that hypothesis 1 gives a simple, precise definition while hypothesis 3 adds arguments to hypothesis 1 without changing the end result. Hypothesis 3 is rejected by Occam's razor.

On syntax: the answers have been out for a while. You should look at the conversions and see it will be impossible to deny they're equivalent. On the basis of this understanding you should be able to see how the AS, AAS, PAAS series works ‒ they all work the same way essentially. This is one of the central concepts of the paper. For a quick reference you should compare figure 11, 12, 13, 14 and 15. What you need to see is that the purpose of each of the three syntaxes is to give you an instruction of how to draw the same tree or graph which is the semantic map (what you would probably call PAS). I'm going to make PowerPoint animations with one-on-one conversions later this year but I'm not sure when.