Rejecting Iteration

This is the first of two chapters exploring the option of resolving various Tolerance Puzzles by denying Iteration, the claim that whatever is possibly possible is possible. In this chapter we grant for the sake of argument that Iteration fails for metaphysical necessity, and consider whether there are other Tolerance Puzzles which remain problematic even on that assumption. Our main focus is on puzzles involving ancestral metaphysical possibility—the status of being either possible, or possibly possible, or possibly possibly possible, or…—for which Iteration is guaranteed by our basic modal logic. We argue that plausible higher-order identities suggest that ancestral metaphysical possibility is not a trivial status even for those who deny Iteration for metaphysical possibility.