Turing machines can't use third order logic. That might be a limitation of the programmer, or it might be a limitation of deterministic machines. A Turing machine can't ever compute anything about it's own output; third order logic might allow deterministic machines which can compute the results of their own output.

I suspect that there is a halting problem where the halting status of machines which only use second order logic cannot be computed in third order logic, just like limits cannot be computed in Peano arithmetic.