Contraries's definition and vacuous truth

Question

Say,

$A: \text{Every Americans use English.}$ $B: \text{No American uses English.}$

$A$ and $B$ are said contarary.

People say that A and B are contrary when

A and B can not be both true but

A or B can be true exclusively or

A and B are both false.

($\uparrow$ D)

But according to the vacuous truth statements, when we assume a possible world(C) where thee are no Americans at all, A and B can both be true!??

What is wrong with my assumption(C) or the definition (D)?

Answer 1

See the Square of opposition.

The relation "being contrary of" is defined in traditional logic for Categorical propositions :

'Contrary' (medieval: contrariae) statements, are such that both cannot at the same time be true.

The issue is with the so-called problem of existential import :

"all $S$ are $P$" implicitly assumes that there are $S$'s in the domain.

In contrast, according to the modern squares of opposition, a view introduced in the 19th century by George Boole, universal claims lack existential import :

In the modern square of opposition, A and O claims are contradictories, as are E and I, but all other forms of opposition cease to hold; there are no contraries, subcontraries, or subalterns.