In epistemic logic, □A is usually interpreted to mean “A is known”, but the difference with “A is certain” is immaterial to the problem with ~□~ as epistemic possibility. The problem is that knowledge is not closed under entailment, so while something can be known its logical consequences may not be, see SEP, Epistemic Closure. For instance, no one knew that the Last Fermat Theorem is true until 1995, when Wiles proved it, even though it is a logical consequence of mathematical axioms that many people knew long before. And idealizing away from constraints on entailed knowledge creates the problem of logical omniscience.
The effect on expressing epistemic possibility is discussed in IEP, Epistemic Modality, which gives a simple counterexample to defining it by negation:
A difficulty for this sort of view involves a proposition that is intuitively ruled out by what someone knows, even though that person does not explicitly know the proposition’s negation. Suppose, for example, that Holmes knows that Adler has stolen his pipe. Holmes is perfectly capable of deducing from this that someone stole his pipe, but he has not bothered to do so. So, Holmes has not formed the belief that someone stole his pipe. As a result, he does not know that someone stole the pipe. According to (Negation), then, it is still epistemically possible for Holmes that no one stole the pipe (that is, that it is not the case that someone stole the pipe), even though it is not epistemically possible for Holmes that Adler did not steal the pipe. This is problematic, as knowing that Adler stole the pipe seems sufficient to rule out that no one stole the pipe, as the former obviously entails the falsehood of the latter. So, S’s not knowing not-p is not sufficient for p to be epistemically possible for S.
An obvious fix is to 'close' knowledge under entailment 'by hand', i.e. to include logical consequences of what is known into constraints on epistemic possibility. In other words, A is epistemically possible when the totality of what is known does not entail ~A. However, this fix creates an opposite problem, it asks for too much.
This resolves the problem in the Holmes case, as Holmes knows something (that Adler stole the pipe) which entails that someone stole the pipe. So, it is not epistemically possible for Holmes that no one stole the pipe, regardless of whether Holmes has formed the belief that someone stole the pipe.
However, views like (Entailment) face problems involving logically and metaphysically necessary propositions. On the assumption that logically and metaphysically necessary propositions are entailed by any body of information, their negations will be epistemically impossible for any subject on this kind of view. If Goldbach’s conjecture is false, for example, then any subject’s knowledge entails the negation of Goldbach’s conjecture. Nevertheless, it is epistemically possible for many subjects that Goldbach’s conjecture is true. So, S not knowing anything that entails not-p cannot be necessary for p to be epistemically possible for S.
Another potential problem is that requiring the entailment of not-p to rule out p seems to result in too many epistemic possibilities. For example, if the detective knows that fingerprints matching the butler’s were found on the gun that killed the victim, that powder burns were found on the butler’s hands, that reliable witnesses testified that the butler had the only key to the room where the body was found, and that there is surveillance footage that shows the butler committing the murder, this would still be insufficient to rule out the butler’s innocence according to (Entailment), since none of these facts properly entail that the butler is guilty.
IEP discusses also some weakenings of the entailment requirement that involve probability. Other fixes are possible along the lines of limiting entailment to what can be deduced without too much non-triviality, the so-called surface information, see What is the difference between depth and surface information?
