Krdytkyu

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)?

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.