I do realize that logic is based on a worldview, namely empiricism. The other day, I ran into some guy telling me to prove reality and argued against the existence of the world. So yeah, there are some people that just deny empiricism as everything was only a perception to them, thanks Nietzsche!
When I am outside of that world, that doesn't make sense.
Not quite. I can very much talk about worlds that do not apply to me, ie. abstract worlds, metaphysical entities, possible worlds etc. For example, an atheist can discuss the capacity of God's omnipotence even if the atheist does not subscribe to the nature of God.
Returning to my p = p example, I do not know of any ideology that denies an object of its own identity, which is why I claim p = p to be an objective fact. Even supposing the denial of the existence of p in the first place, well then the empty set is still equal to the empty set.
You can talk about them, sure. But the things you say will not make sense for those inside those other words. In that sense it is nonsense. It is non-sensical when you want to proof the truth of that world, because truth is local to your world.
Have you ever looked into speculative realism? I'd think they have problems with p=p.
Sorry, I'm not following. How are my claims about a non-local world automatically nonsensical?
Your claims only make sense in your world. They don't make any sense (and thus are non-sensical) to those not in your world.
If you're really interested in this, and also with your logic background, you might want to read Alain Badiou's 'Logic of Worlds' which I think is a nice take on this. Although I don't agree with his finer points, but he does make this point very nicely, I think.
Also, don't thank Nietzsche for that. Nietzsche is one of the most misunderstood and misappropriated thinkers I know of. I don't think he is to blame for what other people have done with his work.