[tel]

Quantifying over T is probably not going to work. In informal terms that reads like "No logic exists where P is independent", which probably wasn't quite what you wanted, but also we can trivially disprove that with T = {}. As long as P is self-consistent, then "not P" should be too.

We're interested in a proposition's status with respect to some theory that we enjoy (i.e. Zermelo–Fraenkel set theory).

[alok-g]

I intended to say the opposite, i.e., for all T (not equal to P or not-P), P is independent, but perhaps that is wrong too.

[tel]

The quantification over T is still kind of weird, though. In a formulation like `for all T, (T and P consistent and T and neg P consistent)` is trivially false, just take `T = {neg P}` and now `{P, neg P}` is inconsistent.

We're never trying to show P is independent of all theories, just some specific one.