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I'm currently debating with someone whether something could be considered possible because we have no evidence that it's impossible. He made the statement that: "All things are possible which are not proven to be impossible", and attempted to justify it by claiming it's a truism (everything that is not ~A is A).
I replied that his statement is not a truism because it doesn't have A/~A as his options, but A and "proven to be ~A".

He then offered me this:

"That objection doesn't seem to hold much merit given that the sentence structure indicates that two concepts are being included. First, you have the two clauses "A and ~A" then you have the contextual modifier "proven to be."

The arguments:

1) ∀X which are ~~ A -->A
2) ~W for ~A -->A

and:

∀X ~W ~A ---> A

Are logically equivalent."

Since I'm new to logical notation, I'm having a hard time expressing the above arguments in normal language.

Can anyone help?

I'm not sure what the notation you quote is trying to achieve. It would help to point out that when speaking of something being possible, there are many different kinds of possibility. At the very least there is:

1. Physical possibility. Things that are compatible with our best understanding of the laws of nature. In this sense, a perpetual motion machine is impossible, meaning that if our understanding of the physics of our universe is correct then none can exist in our universe.




2. Actual possibility. Things that are compatible with the laws of nature together with some statement of the boundary conditions of the actual universe. It is not physically impossible for our solar system to have had seven planets rather than eight, but given the actual boundary conditions, eight is what we get. It is not physically impossible for unicorns to have evolved on our planet, but they didn't.




3. Epistemic possibility. Things that are compatible with what we know to be true, or things that might be true for all we know. This is a much broader kind of possibility than the previous two, because we don't know all the laws of nature or all the boundary conditions of the actual universe. For all we know, Arcturus might have twenty planets in orbit around it, so it is possible in this sense, though with future discoveries we might find that there are ten or a hundred or none. We tend to grade propositions by how credible they are, so we might still say of an epistemically possible proposition that it is highly unlikely.




4. Logical possibility. Roughly speaking, things that do not imply a contradiction. This is also a very broad understanding of possible. It is logically possible that there are unicorns, perpetual motion machines and honest politicians, though there may be none in our universe.

Your disputant appears to have either the third or fourth of these in mind. If some proposition is not proven to be impossible then it is epistemically possible: i.e. it might be true for all we can prove. And if there is no way to prove it false, i.e. there is no way to demonstrate that a contradiction follows from it then it is logically possible.

I think it's important to get clear on what kind of possibility you're talking about. On one way of understanding the kind possibility in question, your friend's claim is trivially true. On another, it's false. Take for example epistemic possibility, which is expressed in statements such as "for all we know, X". On this interpretation your friend's claim is: if we haven't proven ~X, then for all we know, X. This is trivially true we since "we haven't proven ~X" is just taken to mean "for all we know, X".

But if we take "possibility" to be a more objective kind of possibility, then your friend's claim is false. As of now, no one has proven Goldbach's conjecture nor has anyone proven its negation. But, at least on commonly accepted views, Goldbach's conjecture is either necessarily true or necessarily false. (To take another example: it was always necessarily true that water is H2O, so it was never possible that water wasn't H2O, even before this was proven.) Take epistemic possibility again: it's true that "for all we know, Goldbach's conjecture is true" and also true that "for all we know, Goldbach's conjecture is false", since no one has proven either. You and your friend may just have different notions of possibility in mind.