Basically, he argues that there are certain “locating beliefs” (things like “I am John Perry”, “Now is noon,” “Here is the trail that leads out of the woods”) that are logically ineliminable. No matter how you try, you need an indexical in there somewhere to capture them. In your terms, a 0th-person logic could never incorporate these types of beliefs.
This seems closely related to John Perry’s “Problem of the essential indexical” (although it’s been like ten years since I read it): https://dl.booksee.org/foreignfiction/581000/4897fc2fba1f8af4ea7db3d9654bbbb3.pdf/_as/%255Bperry_john%255D_the_problem_of_the_essential_indexica%28booksee.org%29.pdf
Basically, he argues that there are certain “locating beliefs” (things like “I am John Perry”, “Now is noon,” “Here is the trail that leads out of the woods”) that are logically ineliminable. No matter how you try, you need an indexical in there somewhere to capture them. In your terms, a 0th-person logic could never incorporate these types of beliefs.