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What is the difference between proof of inexistence (A) and proof of impossibility of existence (B)? Does A implies B? Does B implies A? I know that there is a scientific axiom that says 'proof lies in the assertion', is the assertion of inexistence scientifically valid?

A little time ago, I read a text saying that, if the scope of the search is reasonable, one can, after experimental examination, affirm the inexistence. Is that so? I ask because I'm not sure if I understood that subject well enough. The text used the example of searching a ball in a house and searching the same specific ball in the universe.

Right now, I think that the difference between A and B is that B implies A and also states the necessity of inexistence, while A only says that right now something doesn't exist, but the possibility that it could have existed in the past or can become existent is open. Am I missing something?

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For the "axiom" I think you mean that the burden of proof lies in the person making the assertion. I don't know if axiom is the right word for that, its more of a pragmatic principle, since it makes science more conservative in what it considers a fact. However, there's no logical reason why you cant assert something and require its refutation by others, that's generally called a working assumption or rebuttable assumption, which is also used in science.

I think you generally have A and B correct. A regards a contingent fact, B is about a necessary fact. The difference is that if B is provable about an object X, then "X exists" implies a contradiction, whereas it does not if it is only A is probable about X. You have noticed that both are proveable if. B by logic alone, while A requires a limited scope of existance, i.e. X can only be in a limited region of of the universe.

For the ball example, not finding the ball in the house, call it ball X, implies it does not exist because we can make the following statement: If X exists then it is in the house. Since we can refute "X is in the house" we can infer "X does not exist".

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Let's start with some logical distinctions. Consider two different sentences, A and B.

A: "X does not exist"

B: "It is impossible that X exist."

Now A and B are very different sentences. A can be false, but B true. Consider the case when X = "Abraham Lincoln". It is clearly true (today in 2014) that "Abraham Lincoln does not exist." However it is just as clearly false that "It is impossible that Abraham Lincoln exist" because at one point in time he did actually exist and everything actual is possible.

But your question was specifically about provability. There are two things to note about the logical relationships between A and B. As I've just shown above, A can be true but B false, so clearly we can't infer a claim like B from a claim like A just in general. However, we can infer a claim like A from a claim like B. Consider the case of a round square. It is impossible for there to be a round square--there simply isn't anything that could possibly be both round and a square. From the fact that it is impossible for round square to exist we can infer that there are no round squares actually.

A separate question is how you would go about proving claims like A or B? The answer there depends upon what kind of object X you are talking about. Suppose we are talking about unicorns. You could hypothetically prove a claim that there aren't any unicorns by simply going and looking. If you could go and examine every object in the universe and find that none of them were unicorns, then you would have proven that there are no unicorns. (Of course, checking every object in the universe for unicornhood is impractical, so most of the time we'd simply use statistics to gauge how sure we can be that there aren't unicorns based on the size of the sample of things we have investigated. So we wouldn't say we've proven there are no unicorns if we've only investigated, but we might have a very, very high degree of confidence in our conclusion that there aren't unicorns even if we haven't strictly speaking proven it.)

But consider the corresponding B claim "It is impossible that there be unicorns." That claim could not be proven to be true or false simply by testing every object in the universe for unicornhood, even if it were practically possible to do so. This isn't to say that no claims at all like B are provable. What would be required to prove a claim like B is to show that the item X somehow involves a logical contradiction. In general the only way kinds of things about which such demonstrations are possible are mathematical objects like The Greatest Prime Number. If you understand the proof correctly you see not just that there doesn't happen to be a greatest prime number, but that in fact there could not be one.

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