Because you can re-define the problem by assuming P finishes if and only if it does not encounter a paradox. It's not hard to see it as possible, as we're already assuming P exists and finishes: allowing it more flexibility should only make it more likely to exist.
And now P is a nonsolution to the original problem (the halting problem). The original problem was to come up with a program that always decides whether its input program halts. We already can build programs that identify the programs which halt and say nothing about programs that don't -- this is a trivial extension to any working interpreter for the language in question.
And now P is a nonsolution to the original problem (the halting problem). The original problem was to come up with a program that always decides whether its input program halts. We already can build programs that identify the programs which halt and say nothing about programs that don't -- this is a trivial extension to any working interpreter for the language in question.