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)”.

Interpretation

Semantically, 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).

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
hı	which X?
ja	λX

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.

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 Nẻo tu bỉo sa tỏqfua means "every cup is on some table", where possibly each cup is on its own table. Certainly, presenting sentences such as $\forall b,\exists t:N(b,t)$ and assigning them this interpretation is popular in discussions of singular predicate logic.

But because the plural-logic tu bỉo ranges over all "cups-es", i.e. all groups of cups, it also includes the referent "bío = all relevant cups together". Consequently, this sentence ends up saying that all cups are on the same table, namely whichever sa tỏqfua this maximal bío is on.

As a concrete demonstration, if there are three cups B1, B2, B3, then Nẻo tu bỉo sa tỏqfua claims all of the following:

• There is some table that B1 is on.
• There is some table that B2 is on.
• There is some table that B3 is on.
• There is some table that [B1 roı B2] are on.
• There is some table that [B1 roı B3] are on.
• There is some table that [B2 roı B3] are on.
• There is some table that [B1 roı B2 roı B3] are on. (So they are all on the same table!)

You can explicitly quantify over single cups to get the intended meaning: Nẻo tushı bỉo sa tỏqfua.