> they can read and write

Note that I’m not joining people with AND. I joined abilities.

For instance, to expand on the example.

I have three Pokémons. They can read and write.

I’m not saying that my three Pokémon are the intersection of reading Pokémon and writing Pokémon. There might be other Pokémon who can do just the same.

I’m saying that what they can do is a union of writing and reading.

EDIT: spelling

But they are in the intersection of "entities that can read" with "entities that can write".

Yes having attributes containing the union of the sets of attributes {“can read”} and {“can write”} means being in the intersection of the set of things having the attribute “can read” and that having the attribute “can write”. Set union, intersection, and membership are related that way.

(Which means saying that “and” is about intersection and not union is nonsense, everything statement about set union can be rephrased to an equivalent statement about intersection, and vice versa.)

Yeah, there's some duality there. If each predicate is identified with the set of things that satisfy that predicate, then the "and" of two predicates is identified with the intersection of the sets that satisfy the two predicated. On the other hand, if you instead use predicates by talking about the set of predicates that objects satisfy, then saying that something satisfies one set of predicates, and satisfies some other set of predicates, then that says that the thing satisfies all the predicates in the union of those two sets.

For some relation R between elements of the set X and elements of the set Y (we can think of X as being the set of individuals, and Y as being a set of elementary unary predicates), for each x in X, we can refer to A(x) := {y in Y : x R y} , and for each y in Y we can talk about B(y) := {x in X : x R y} . When we say (x_1 R y_1) && (x_1 R y_2), this is equivalent to both (x_1 in B(y_1)) && (x_1 in B(y_2)) i.e. (x_1 in (B(Y_1) intersect(B(y_2)))) , and also to (y_1 in A(x_1)) && (y_2 in A(x_1)) , i.e. ({y_1} \subset A(x_1)) && ({y_2} \subset A(x_1)) i.e. (({y_1} union {y_2}) \subset A(x_1)) .

Personally, if we are talking about a single x_1, and we are saying "and" about a number of different y_1 , y_2 , etc. , or if we are talking about each x in some collection but are saying the same thing about each of them, namely that for each x in the collection, that x R y_1 and that x R y_2 and that x R y_3 , I think it seems more natural to take the first of the two framings described above. That is, I think it seems more natural to say that for each x in the collection, that x \in (B(y_1) intersect B(y_2) intersect B(y_3)), rather than saying that ({y_1} union {y_2} union {y_3}) subset A(x) .

Or, if we give the name W to the collection we are talking about, we could say that "W subset (B(y_1) intersect B(y_2) intersect B(y_3))". This seems to me a fairly natural way to say "all the W , y_1 , y_2, and y_3" ("each of my 3 pokemon can do reading, writing, and arithmetic") .

The other way would end up more like, "({y_1} union {y_2} union {y_3}) subset (the intersection over all x in W, of A(x))".

This works, but seems less natural to me.

Also, especially in the case where you expect X to be substantially larger than Y.

"I have three Pokémons. They can read and write." cf: I have three Pokémons that can read and write.

Now, thanks to what I can only describe as the implicit vagaries of English, the bounding (three) might now imply that I have three ... or more Pokémons! To make that sentence precise: I have three Pokémons and they can read and write. Here I am substituting and for a full stop (period.) When I deployed "that" I managed to muddy the waters.

Mapping logical and Boolean operators to grammatical constructs is tricky and should not be taken lightly. Best to consider them separately. AND is not and (except when it is!)

Be careful out there kids.

It depends if you consider the implied set to be the people (of which there are readers and writers) or the capabilities (reading and writing).