I believe that an argument can "exist", so to speak, without any humans believing in it. Logic is not suddenly invalided by the technicality of nobody following it. To state that it is impossible to show my my assertion is logically invalid means just that: It cannot be disproved.
You have a narrow minded view of logic.
Logic is only the technical form of reasoning, given two or more premises, from one fact to another.
The premises must, however, be true for the conclusion to be valid.
If I were to say: "monkeys are one inch tall, pink, and smell like terpintine." I can form a perfectly valid set of logical reasoning based on that premise, but it is inevitably a waste of time, because any conclusion I draw from a premise which is absolutely false will be invalid. The conclusion may still be true, but that would not be because you have put forward a rational and consistent argument based on true premises. It would be merely co-incidence.
So, your logic may be perfectly valid, if it is encapuslated in your mind and you have never spoken a word of it, but it does not make the conclusions valid. Conclusions are only valid, and of absolute truth, when the premises are valid. One of the premises or 'assumptions' you make when you begin an argument is that you are arguing with someone. If you are not then your conclusion is invalid.
To recap:
When you put forward an argument, you attach to the argument a number of additional premises which are understood but not explicitly stated. Those premises you may refer to as assumptions, but the are necessarily premises of the logical argument as well.
These are premises such as:
You speak english, I speak english, and we both understand eachother.
We are both able to communicate in a meaningful way.
Truth is absolute and therefore there is a reason to be arguing in the first place.
I am not simply a figment of your imagination because arguing with yourself is irrational, as arguments are debates between two or more people.
We both exist, and are both alive, and are both conscious, and both have free will.
And so on.
Whenever you actually take your argument from your thoughts and turn them into language, that is: start a debate. You attach all of these premises (and more) to your argument. It is perfectly valid for me to point out that your argument contradicts an implicit premise.
And this is actually entirely essential to epistemology; the study of knowledge.
To argue outside of this scope is to argue irrationally and without the essential motive to determine truth from falsehood; knowledge from opinion.