Implied existence

The short answer is "yes."
Now, then, what has led you to consider that it might not?

A does not believe B

The key is 'does not'. A doesnt have to exist, because there is no 'believing' and hence need not be a 'believer'.

I disagree on the grounds that the negative neither denies neither the existence of the action, (but, instead, directs the action towards the contrary of it's positive direction,) nor does it deny the subject, which continues to 'be' in action, only in the negative, not the positive sense.

It might be argued that the negative form; i.e., "does not," denies the existence of the actio

Is this thread a bit crazy or I got the wrong impression?

- "I don't believe you."

Perhaps I don't exist because I don't believe you?

- "He doesn't work on Mondays."

He doesn't exist at all? Or perhaps he exists from Tuesday through Sunday, and he stops existing on Monday only to be reborn again the following Tuesday?

It's also possible tha

Whether something is implied or not depends entirely on the intent of the person who wrote/uttered the phrase. The more appropriate question would be whether the existence of A could be inferred, in which case the answer depends on whoever is interperting the phrase and whether or not they chose to infer the existence of A.