- The mediocrity principle is true
The mediocrity principle is a heuristic not a hard and fast rule. And it more or less states the obvious that is that rare examples are rare and that if you took a 1 shot approach of picking from a set of variables, it's more likely that you've picked a common one. That tells you nothing about whether or not your choice is rare and it definitely doesn't rule out that rare choices do exist.
But following this principle works in the majority of cases so it's a good rule of thumb it's not suitable to be used in such arguments though, as what follows from it is at best again a heuristic not a certainty.
- If the mediocrity principle is true then we're not "special, privileged, exceptional, and even superior"
We shouldn't assume we are, but we still could be. Also it reduces the general applicability of rules quite significantly if all you deal with are edge cases, so yeah it makes sense to not assume you're an edge case until that is proven to be the case by more observations. So again good heuristic not a rule.
- We're not "special, privileged, exceptional, and even superior" [1, 2 modus ponens]
Can't deduce with certainty from a heuristic, so no that does not follow.
- If we're not "special, privileged, exceptional, and even superior" then we're physical and scientific physical laws apply to us
Also the laws of physics aren't hard and fast rules but heuristics based on patterns of observation. Breaking them is not generally impossible and has been done before, however it would likely result in a paradigm shift or would hint at special cases either of the event that broke them or maybe we're just living in an edge case. Either way breaking them would mean that something we took for granted doesn't work that way.
Also we ARE physical beings in the sense that our existence can be detected, measured and quantified by physical means and our bodies can be described by science.
But it's not that physics mandates a determinism, it's rather the other way around, in that it's a good idea to assume one, because if things happen arbitrary and without patterns we're out of luck describing and making sense of them. But science is descriptive not prescriptive, it can make predictions but they are assumptions not certainties, because they are reasoned starting from shaky grounds. So they are internally valid (or could and should be), but it's not certain that they are sound and science doesn't claim otherwise.
- We're physical and scientific physical laws apply to us [3, 4, modus ponens]
We are but that doesn't follow from 3 and 4 but rather from the fact that a description of us by science is somewhat successful.
- If we're physical and scientific physical laws apply to us then determinism is true for us
We'd assume that to be the case and the rag doll physics for our corpses seem to work well, but that's still insufficient to cast a judgement.
- Determinism is true for us [5, 6 modus ponens]
It's not entirely unreasonable in some domains, but insufficient to draw that conclusion.
- If determinism is true for us then we don't have free will
- We don't have free will [7, 8 modus ponens]
Apparently even that wouldn't convince philosophers who invented compatibilism.