The algebra of toki pona phrases

Conlangs Toki Pona
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I was reading @_at's blog post about toki pona noun phrases and, thought that I, in the spirit of @zearen's post about underfilling, would make a similar discussion-oriented post about the semantics of phrases (not just noun phrases, at least also verb phrases) in toki pona.

Abstract algebra

First of all, the reason for the "algebra" in the title is that abstract algebra seems to me to be exactly the tool we need in order to formalize "how things combine together". And it can be very abstract indeed: rooted full binary trees whose leaves are members of a set are the free magma on that set, so any way of parsing a noun phrase into a (rooted and full) binary tree and then composing the elements together into a single value in a way where every internal node uses the same operation can be described algebraically.

However, we probably want something a bit more structured than just a magma. For example, we can consider "pi" as separating noun phrases into two "simple noun phrases", and we could interpret each simple noun phrase as the intersection of the semantic spaces of each of its words. Then, the algebra of simple noun phrases would be both commutative and associative: a commutative semigroup. If we consider "ijo" to be completely non-specific (for the record, I have no idea if this is actually reasonable, it's just for illustration purposes), then we have an identity element, which gets us from a (commutative) semigroup to a (commutative) monoid. Similar considerations apply to how simple noun phrases combine with each other.

In this way, we can think of the type of algebra we want based on the properties it satisfies. Even if we can't use intersection directly (consider "sitelen sitelen" which with intersection is no different from just "sitelen"), we probably still want simple noun phrases to be done with at least a semigroup. We can also go the other way: if ona's nasin distinguishes, say, "soweli jan" and "jan soweli", then we know that we don't have commutativity. As another example, the fact that people avoid multiple "pi" due to perceived ambiguity is an indication that the algebra of how simple noun phrases combine with each other might not be associative!

Special behavior of "ala"

I submit for your consideration an example that, if you consider it grammatical and it can mean what I'm trying to use it to mean, either shows that not even all simple noun phrases are restrictive, or that "ala" needs to be treated differently at the syntax level (which @_at already argued in the case of noun phrases without "pi"): "soweli pi jan ala", for non-person beasts. This time, "jan ala" cannot refer to the set of all "jan", instead (I propose), "ala" is a negation operator in some sense, most obviously but not necessarily set complementation, meaning that "soweli pi jan ala" would refer to the set of soweli which are not jan, which is what we (or at least I) wanted.

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I disagree with your analysis of noun phrases as commutative and associative. To me they have tanru-like semantics. "soweli jan" describes the subset of soweli that are jan-like (including such interpretations "the jan's soweli") and that usually isn't the same as the subset of jan that are soweli-like. Also, although "soweli jan pona" and "soweli pona jan" are both soweli that are jan-like and pona-like, a soweli jan that is pona doesn't have the same connotations as a soweli pona that is jan.

ijo sort of acts like an identity element sometimes, but what about possession? "ijo mi" pretty clearly means something different to "mi". (I guess it's broader, I would accept ijo mi as "the ijo that is me" and then that means the same thing as mi... really the problem is that true identity elements are useless for communication so the implication in using "ijo mi" is that it doesn't just mean "mi" and you used the ijo for a reason)

I agree with the part about ala.

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Yeah, this one I have already been told, I went and asked some questions in the beginner questions thread after posting this. I didn't update this post because I intended more as encouraging a general discussion about this sort of analysis than me trying to propose any particular analysis.

This, however, is at least partially news to me. Would you say that they differ other than in connotation? Because I was under the strong impression that the modifiers at least commuted, at least in denotation (whether in focus is a different thing).

Also, for the record, I don't think you've disagreed that it's associative? Isn't that's what "pi" is for (if you see that as syntax, which I do genuinely do for now), to forcefully regroup the string? Giving the result that noun phrases containing "pi" are not associative, but the two strings on either side are.

EDIT: maybe a better way to think of it would be that they're not associative, but the syntax enforces left associativity except for pi? So "X Y Z pi A B C" is syntactically "(((X Y) Z) ((A B) C))"

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Would you say that they differ other than in connotation?

I personally think they can, though I don't know where the line between denotation and connotation is. That seems like a hard thing to distinguish in Toki Pona in general..

I was about to ask what you meant by "the two strings on either side [being associative]" but your edit describes my mental model of noun phrases pretty exactly. "tomo sona meli lili pona" is a "(pretty (little (girls' school)))". In particular, if something can be described by the noun phrase x_0...x_n where all x_i are content words (or pi phrases, but with all of the pi phrases following all of the content words), then it can also be described by x_0...x_{n-1}. Any tomo sona meli is also a tomo sona, and so on.

However, a x_0...x_n doesn't have to be x_n in the conventional sense of x_n, just relative to the usual values of x_0...x_{n-1}. I'd describe a dark grey as pimeja walo, even though dark grey is not usually walo.

Here's a contrived theoretical example of non-commutativity in modifiers: Imagine a world inhabited by bugs where it makes sense to distinguish between dark and light bugs (maybe they're two different species whose most striking difference is color). Now imagine that in both species there is some very slight color variation, so there are pipi pimeja who are dark grey and pipi walo who are light grey, instead of pitch black and white. It'd make sense to me to describe them as pipi pimeja walo and pipi walo pimeja respectively. I think if you said "pipi walo pimeja li kama" and you were talking about a dark bug who is a little grey it would be very confusing at best, maybe even a false statement.

(That kind of thing happens often in practice. A new teacher is a "jan sona sin", and a knowledgeable newcomer is a "jan sin sona", and even though in this case both of them are sona and sin in the conventional senses, it would be weird to swap them around.)

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Seems like your link broke :sweat_smile:, here is the original post: [2/100] toki pona and the semantic subset | DK’S ABODE

I think that it is accurate to say jan ala in the context above cannot share a referent with jan. This is an interesting counter-example to the argument I present at the bottom of the OG post (copied here for posterity):

I contest that \operatorname{ref}(\text{jan ala}) is equivalent to \operatorname{ref}(\text{jan}) in this context. This sentence makes the claim (\forall p \in \text{People})\, p \text{ will not have fun}. The direct negation of this preposition is (\exists p \in \text{People})\, p \text{ will have fun}. This corresponds to the toki pona sentence "jan li kama musi" (People will start having fun), which is the direct negation of "jan ala li kama musi". Vitally, using "jan ala" as a subject or a direct object makes a logical claim spanning all people. Therefore, \operatorname{ref}(\text{jan ala}) is the set of all people and must be equivalent to \operatorname{ref}(\text{jan})

There are two other situations I can think of where ala makes a claim across all of noun phrase's referents, as long as it's on the outside of the noun phrase:

  • In a la context clause: soweli ala la ona li moku e soweli // None of the animals eat meat I'm not sure this one works so well
  • In the e direct object: soweli li moku e kala ala // The animals don't eat any fish

I think it's totally possible these are the only cases in which ala works like this. I agree with the special case interpretation.

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I see tp phrases as usually having an implicit "of" when you apply a modifier.

e.g.
soweli jan pona = (soweli of jan) of pona
soweli pona jan = (soweli of pona) of jan
soweli pi jan pona = soweli of (jan of pona)

Even an adjectivial modifer like in "jan pona" (person that is good) can be analysed this way, using the action noun of the adjective (person of goodness).

To me this is analogous to an implicit noncommuntative multiplication operation, like in matrix algebra. e.g. A = XB.

And I see "ale" and "ala" on noun phrases as affecting quantification.
jan ale li pona.
∀j, jan(j) ⇒ pona(j)

jan ala li pona
∄j, jan(j) ⇒ pona(j)

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how about a non-distributive sentence like "jan ale lon li kama kulupu" - "all the people here became a group"?

it's true that all of the people together became a group, but it doesn't make sense to say that each individual person became a group, i.e. ∀j, jan(j) ⇒ kama_kulupu(j) doesn't accurately describe this sentence

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