 Clearly we have a family of grue predicates in this problem, one for each different “critical time” \(t\); let’s write grue\((t)\) to denote these grue predicates. Following Goodman, let us refer to methods as  projection rules  in discussing this example. A projection rule succeeds in a world just in case it settles on a generalization that is correct in that world. Thus in a world in which all examined emeralds are found to be green, we want our projection rule to converge to the proposition that all emeralds are green. If all examined emeralds are grue\((t)\), we want our projection rule to converge to the proposition that all emeralds are grue\((t)\). Note that this stipulation treats green and grue predicates completely on a par, with no bias towards either. As before, let us consider two rules: the  natural projection rule which conjectures that all emeralds are green as long as only green emeralds are found, and the gruesome rule which keeps projecting the next grue predicate consistent with the available evidence. Expressed in the green-blue vocabulary, the gruesome projection rule conjectures that after observing some number of \(n\) green emeralds, all future ones will be blue. The figures below illustrates the possible observation sequences, the natural projection rule, and the gruesome projection rule.