Being extremely pedantic, I will frequently require those that I discuss or debate with to adhere to the rules of debate. A brief introduction to these rules is available as a faq for alt.atheism. However, I will briefly go over a couple of the most commonly occuring points.
In logical debate, the burden of proof is always upon the person making the positive assertion. This principle is rather simple, but also rather deceptive. There exists a standard formation of a question to determine whether or not the proposition is indeed a positive assertion. As a common example, many people claim that those who claim that gods do not exist have the burden of proof, just as much in fact as those who claim that gods do exist. First of all, it should be perfectly clear to all that those who claim that "gods exist" have the burden of proof. However, those who claim that "gods do not exist" are in fact making an assertion, but a negative one. The standard formation of the assertion is Not There Exists gods. From this formation, it becomes clear that although it is indeed an assertion, it is not a positive assertion and does not in argument have the full burden of proof. However, the burden of proof may be properly shifted to such a person however if a prima facie case is established, which brings us to the next point.
Some people do not really understand the why on that last point, so I shall attempt to explain further. The reason that a negative claim does not have the full burden of proof is because of the fact that they are claiming something to be false. To prove that in science is nearly impossible. While that hardly excuses a proposition, it is however a form of default position. If one assumes that things are false until shown otherwise, one is not likely to believe a positive assertion without reason, and that is part of the point of having the burden of proof--to avoid believing something is established when it has not yet been so. However, one is in danger of believing something false that is true, for this reason, there is some burden of proof on the belief in the negative. Again, the burden is to establish a prima facie case in support of ones position. Once one has done that, then one has established at least a reasonable reason for ones position. The phrase Burden of Proof is deceptive, for it doesn't mean rock solid proof, it means establishing of a rational case in defense of the position.
If one rejects a claim, one is not necessarily holding it to be false. One is merely saying that the claim is not true. One is in fact making no statement whatsoever about if the claim is false or not. To claim otherwise is to make an assumption about what the person has said. Assuming something about another's position can be hazardous at best. In logic, there is a axiom, known as the Law of the Excluded Middle. This axiom states roughly that ¬¬P = P. It is important to note that not all people accept this axiom, and indeed, many logicians reject it. One such branch of logic that rejects this axiom is known as Intuitionism.
A similar argument arises when one claims that an entity may exist. There is an extension to predicate logic called modal logic which addresses this very idea. Simply put, it is a claim that must be justified to claim that something is possible. Formally, if we state that some proposition P is possible, then we are claiming that if P is true, then we do not produce a contradiction within our universe. Note that this is a claim, and cannot be assumed. In fact, many arguments stem precisely out of the question of is some P possible. Further, note that just as one cannot assume P is possible, one cannot assume that ¬P is possible, or that P is not possible. Each one is a different statement that cannot be assumed in modal logic.
Formally, modal logic extends predicate logic in the following manner. It adds two new quantifiers, "It is possible that", and "It is necessary that". The symbols used are a diamond for the former, and a square for the latter. If we say It is possible that P, we are stating that either P is satisfyable, or that P is may be true. In other words, we are stating that we do not produce a contradiction in out logical universe if P is true.
The second quantifier, "it is necessary that", is similar in meaning. If one states It is necessary that P, one is stating that P must be true for all possible models.
It cannot be stressed enough, that like universal and existential quantifiers, one cannot assume necessary and possible quantifiers in logic. Nor can one assume the negation.
There are several standard relations within modal logic. Here, I shall attempt to illustrate them here (assuming non-intuitionistic logic):
¬<> ¬P <-> [] P
¬[] ¬P <-> <> P