# Distributivity

## Formal definition

A predicate $P(\mathbf {x} )$ is distributive in terms of its argument $\mathbf {x}$ iff

$\forall \mathbf {x} .P(\mathbf {x} )\iff \forall x\in \mathbf {x} .P(x)$ 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, ${\textsf {dua}}(x,y)$ 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 nondistributive because each of two things are not themselves two.

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.

## More examples

• reı is nondistributive because when six snakes surround you, it’s not true that each one of those snakes also surrounds you.