Paradox

Ok so I just got finished watching T2 and got to wondering. If they destroyed the chip before Arnie went into the steel wouldn't he just cease to exist? Think about it, if he came from the future and they destroyed the lab and then the info that started the whole lab to begin with i.e. the arm and the chip then wouldn't it make more sense if he just vanished? If you change the future in the now then if he came from the future that has just been changed then he couldn't exist anymore because they changed it. It's kinda like if you went back how ever many years you are old plus one and killed your dad, then you wouldn't have been concieved and therefore wouldn't exist. So therefore I claim shananigans on Terminator 2: Judgement Day. They created a time paradox. Heck things should've started changing as soon as Niles said he quit. Am I right?