While I'm not stating that one can know what logic is not available to us, that does not mean that it is impossible to deduce a generic case that there exists logic beyond what we know.  That "statement" is well within our logic as here it is, it exists.  If you could see numbers in a group from 1 to 11 and all you knew were 1 to 11, you may not know about 12 or 0 or -5, but that does not mean that you cannot think of the possibility of some number greater than 11 or less than 1.  The logic dealing with the possibility of existence is definitely within our logic base, but that does not mean that we have the ability to know all that it consists of or the means to understand it even if we did know what it consisted of.  Just the aspect that logic could have an infinite amount of subsets of which there is most likely an infinite amount of subsets outside of our collection of subsets shows that we would at least not be capable of knowing all that makes up the logical spectrum since we are mortal and will not have the time or capacity needed to understand an infinite number of subsets in its entirety.

While there may not seem a way to "prove" it as of yet and may never be.  It still is a possibility.  If we knew all that there is to know then there wouldn't be many people searching for solutions to problems.  And you'll have to excuse me, I think my mind is escaping me.   In fact, I'm not entirely sure I caught your whole argument because of the fact, but I think I roughly responded to it.