Distributivity: Difference between revisions

Latest revision as of 16:01, 16 February 2026

In plural logic, a predicate $P$ is distributive over its argument $xx$ if and only if these two statements are equivalent:

$P(xx)$ .
For all $x$ among $xx$ , $P(x)$ .

For example, ${\textsf {dua}}(x,y)$ distributes over both its arguments since if all of a group of people know several things, this means that each of them knows each of those things. Meanwhile, ${\textsf {gu}}(x)$ is non-distributive because each of two things are not themselves two.

Being explicit

Forcing distributivity

If you have a non-distributive verb like hao, how do you distribute it over some plural constant?

One way is to use the word shıcha “▯ satisfy property ▯ individually/one by one,” in a serial verb.

Shıcha hao náq.
The guys hao one by one.

Another is to use tú mea X “each among X”.

Hao tú mea náq.
Each of the guys haos.

Forcing non-distributivity

If you have a distributive verb like geı, how do you apply it to a plural subject without getting a distributive meaning?

😕 Geı ké deo gu tíefuq.
The two children (each) wear the coat.
Both D₁ wears T and D₂ wears T — nonsense.

One way is to use the word mecha “▯ satisfy property ▯ as a group,” in a serial verb.

Mecha geı ké deo gu tíefuq.
The two children wear the coat together.
(D₁+D₂) considered as a unit wears T

Another is to use me “an aggregate of….” An aggregate is a plural value wrapped in a singularizing wrapper, shielding it from effects like distributivity.

Geı ké me kê deo gu tíefuq.
The two-child-bundle wears the coat.
(D₁+D₂) considered as a unit wears T

@@ Line 1: / Line 1: @@
-== Formal definition ==
+In [[plural logic]], a predicate <math>P</math> is '''distributive''' over its argument <math>xx</math> if and only if these two statements are equivalent:
-A predicate <math>P(\mathbf x)</math> is '''distributive''' in terms of its argument <math>\mathbf x</math> iff
-: <math>\forall \mathbf x. P(\mathbf x) \iff \forall x \in \mathbf x. P(x)</math>
-or, in English, iff
-: for any given plural constant, to say that it satisfies P must mean the same as saying that each singular member of it also satisfies P.
-For example, <math>\textsf{dua}(x, y)</math> ''distributes'' over both its arguments since if all of a group of people know several things, this means that each of them knows each of those things. Meanwhile, <math>\textsf{gu}(x)</math> is nondistributive because each of two things are not themselves two.
+# <math>P(xx)</math>.
+# For all <math>x</math> among <math>xx</math>, <math>P(x)</math>.
-Another way to think of distributivity is that it shifts verbs back into the realm of singular logic, where all statements are meaningful of singular objects. Toaq’s grammar “does not care” about distributivity, since singular logic works in precisely the same way in plural logic contexts, whereas the converse is not true.
+For example, <math>\textsf{dua}(x, y)</math> ''distributes'' over both its arguments since if all of a group of people know several things, this means that each of them knows each of those things. Meanwhile, <math>\textsf{gu}(x)</math> is non-distributive because each of two things are not themselves two.
-== More examples ==
+== Being explicit ==
-* {{t|reı|{{x}} surrounds {{x}}}} is nondistributive because when six snakes surround you, it’s not true that each one of those snakes also surrounds you.
+=== Forcing distributivity ===
+If you have a non-distributive verb like {{t|hao}}, how do you distribute it over some plural constant?
+One way is to use the word {{t|shıcha}} “▯ satisfy property ▯ individually/one by one,” in a [[serial verb]].
+{{example|Shıcha hao náq.|The guys <i>hao</i> one by one.}}
+Another is to use {{t|tú mea X}} “each among X”.
+{{example|Hao tú mea náq.|Each of the guys <i>hao</i>s.}}
+=== Forcing non-distributivity  ===
+If you have a distributive verb like {{t|geı}}, how do you apply it to a plural subject without getting a distributive meaning?
+{{example|😕 {{red|Geı}} ké deo gu tíefuq.|The two children {{red|(each) wear}} the coat.<br><small>Both D₁ wears T and D₂ wears T — nonsense.</small>}}
+One way is to use the word {{t|mecha}} “▯ satisfy property ▯ as a group,” in a serial verb.
+{{example|Mecha geı ké deo gu tíefuq.|The two children wear the coat together.<br><small>(D₁+D₂) considered as a unit wears T</small>}}
+Another is to use {{t|me}} “an aggregate of….” An '''aggregate''' is a plural value wrapped in a singularizing wrapper, shielding it from effects like distributivity.
+{{example|Geı {{green|ké me kê deo gu}} tíefuq.|{{green|The two-child-bundle}} wears the coat.<br><small>(D₁+D₂) considered as a unit wears T</small>}}