Determiner

A determiner is a particle that consumes a verb form and produces a noun form — specifically, a determiner phrase.

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

Determiner particles

Word	Meaning
	X (bound; see below)
sá	some X
tú	every/each single X
tútu	every group of X-es (see below)
túq	all the Xs together
sía	no X
ní	this/that X
hú	endophoric determiner
ké	exophoric determiner
báq	X in general, X-kind
já	λX, see Property
hí	which X?

Semantics

Formally, grammatical determiners tend to correspond to logical quantifiers over a now-bound variable, plus an occurrence 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, sá bıo 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.

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

Every, each, all

tú bıo quantifies over single cups, i.e. each individual cup. This is often what we want to say, despite not being the "purest" form of for-all quantification in plural logic; after all, groups of several cups are also bıo. The expression tútu bıo quantifies over the range of "cups-es": the possible values of bío then include not only individual cups, but also groups of cups.

You can read tútu bıo nä … as: "for all xx, if xx are some cups…"

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