Determiner

A determiner is a particle that consumes a predicate phrase and produces a noun phrase.

For example: sa “some” is a determiner, bỉo “…is a cup” is a predicate phrase, and sa bỉo is a noun phrase meaning “some cup(s)”.

Determiner particles

Word	Meaning
sa	some X
tu	every X
tushı	each X
tuq	all X
sıa	no X
ke	the X
hoı	the aforementioned X
baq	X in general, X-kind
ja	λX
hı	which X?
co	how many X?

Additionally, can be analyzed as a tonal pseudo-determiner that refers to bound variables, or falls back to "implicitly-bound" ke X if there is no earlier binding.

Semantics

Formally, grammatical determiners tend to correspond to logical quantifiers over a now-bound variable, plus an occurence of that variable. For example, the sa determiner corresponds to the $\exists$ quantifier. The tagged predicate phrase doubles both as a domain and a name for the variable.

In short, sa bỉo does three things:

1. introduces an existentially bound variable bío to the clause;
2. specifies that it refers to a cup (or some cups: see plural logic);
3. acts in its place in the sentence as an instance of this variable.

Hẻaq jí sa bỉo.
${\color {brown}\underbrace {\exists {\textsf {bio}}:} _{1}}\;{\color {chocolate}\underbrace {{\textrm {Bio}}({\textsf {bio}})\mathop {\wedge } } _{2}}\;{\textrm {Heaq}}({\textsf {ji}},{\color {teal}\underbrace {\textsf {bio}} _{3}})$
I'm holding some cup(s).

Every, each, all

tu bỉo quantifies over the range of "cups-es". The possible values of bío include not only individual cups, but also groups of cups. A group of cups is also a bỉo, after all.

This can lead to surprising behavior: see below.

tushı bỉo quantifies over "cups-es that are one", i.e. each individual cup. It's like tu bỉo ru shỉ.

tuq bỉo doesn't make a "for-all" statement. Instead it refers to the single entity "all cups (together)".

Inappropriate tu

You might think that Zủdeq tu pỏq sa zủ means "every person speaks some language", where possibly each person speaks a different one. Certainly, presenting sentences such as $\forall x,\exists y:P(x,y)$ and assigning them this interpretation is popular in discussions of singular predicate logic.

But because the plural-logic tu pỏq ranges over all "people-s", i.e. all groups of people, it also includes the referent "all relevant people together". Consequently, this sentence ends up saying that all people speak at least some common language(s), namely whichever sa zủ these maximal póq speak.

As a concrete demonstration, if there are three people P1, P2, P3 in question, then Zủdeq tu pỏq sa zủ claims all of the following:

• There is/are some language(s) Z1, that P1 speaks.
• There is/are some language(s) Z2, that P2 speaks.
• There is/are some language(s) Z3, that P3 speaks.
• There is/are some language(s) Z4, that [P1 roı P2] speak.
• There is/are some language(s) Z5, that [P1 roı P3] speak.
• There is/are some language(s) Z6, that [P2 roı P3] speak.
• There is/are some language(s) Z7, that [P1 roı P2 roı P3] speak. (= a common language! ⚠️)

You can explicitly quantify over single people to get the intended meaning: Zủdeq tushı pỏq sa zủ.