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Grammatically, a '''quantifier''' is a particle that consumes a predicate phrase and produces a noun phrase.
A '''determiner''' is a particle that consumes a predicate phrase and produces a noun phrase.


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


These particles are called “quantifiers” because they semantically corrsepond to a ''logical'' quantifier over a now-bound variable, plus an occurence of that variable. The consumed predicate phrase doubles both as a ''domain'' and a ''name'' for the variable.
Semantically, these particles tend to correspond to logical '''quantifiers''' over a now-bound variable, plus an occurence of that variable. For example, the {{t|sa}} determiner corresponds to the <math>\exists</math> quantifier. The tagged predicate phrase doubles both as a ''domain'' and a ''name'' for the variable.


In short, {{t|sa bỉo}} does three things:
In short, {{t|sa bỉo}} does three things:
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<blockquote>
<blockquote>
{{t|Hẻaq jí <u>sa bỉo</u>.}}<br>
{{t|Hẻaq jí <u>sa bỉo</u>.}}<br>
<math>{\color{red}\exists \textsf{bio}}: {\color{orange}\textrm{Bio}(\textsf{bio}) \,\wedge} \, \textrm{Heaq}(\textsf{ji}, {\color{teal}\textsf{bio}})</math>
<math>{\color{red}\exists \textsf{bio}}: {\color{orange}\textrm{Bio}(\textsf{bio}) \,\wedge} \, \textrm{Heaq}(\textsf{ji}, {\color{teal}\textsf{bio}})</math><br>
I'm holding some cup(s).
</blockquote>
</blockquote>
== Determiner particles ==
{| class="wikitable"
|-
! Word !! Meaning
|-
| {{t|sa}} || some X
|-
| {{t|tuq}} || every X
|-
| {{t|tushı}} || each X
|-
| {{t|tuq}} || all X
|-
| {{t|sıa}} || no X
|-
| {{t|ke}} || the X
|-
| {{t|hoı}} || the aforementioned X
|-
| {{t|baq}} || X in general, X-[[kind]]
|-
| {{t|hı}} || which X?
|-
| {{t|ja}} || λX
|}
Additionally, {{tone|2}} can be analyzed as a tonal pseudo-determiner that refers to bound variables, or falls back to "implicitly-bound" {{t|ke X}} if there is no earlier binding.
=== Every, each, all ===
{{t|tu bỉo}} quantifies over the range of "cups-es". The possible values of {{t|bío}} include not only individual cups, but also groups of cups. A group of cups is also a {{t|bỉo}}, after all.
This can lead to surprising behavior (TODO example), and you want to say {{t|tushı}} instead.
{{t|tushı bỉo}} quantifies over "cups-es that are one", i.e. '''each''' individual cup. It's like {{t|tu bỉo ru shỉ}}.
{{t|tuq bỉo}} doesn't make a "for-all" statement. Instead it refers to the single entity "all cups (together)".