OWL is built on the open world assumption, and you can't really get around that. In some cases, you can provide some additional axioms about a class to reduce the number of possibilities that an open world permits. E.g., if you have
{d} ⊑ ∃ hasValue exactly 1 {a, b, c}
a ≠ b
a ≠ c
b ≠ c
then from hasValue(d,a), you could infer both of these:
¬hasValue(d,b)
¬hasValue(d,c)
What you're asking for sounds more like default reasoning where you're trying to say that
If X isn't provable using standard OWL, then infer ¬X.
The difficulty in this is that you can represent both positive and negative statements in OWL, and you'd have a problem when neither X not ¬X is provable using standard OWL, since in that case, you'd infer both ¬X and ¬¬X.
However, there are some things that you can do here. If you're generating the data, you can add negative property assertions to explicitly state that some particular relation doesn't hold. That's one way of encoding total knowledge about some subset of your domain. Thus, if there are a number of known B's out there, then you can assert for each one whether an A is related to each one or not.
Another option would be using SPARQL to query the data and check whether an A is related to a qualifying B, and if it's not, then add some additional data that will an OWL reasoner draw the inferences that you need. This isn't too hard, but if any of the other relations that need to find the B's that you're looking for are inferred, you still might need an OWL reasoner running prior to the SPARQL queries.