where X and the Xi are members of some set S. The premises are X1, X2, . . . , Xn and the conclusion is X. We use such rule instances to specify an inductively defined subset A ⊆ S; intuitively, the rule (1) says that if you have X1, . . . , Xn ∈ A, then you must also take X ∈ A. Note that since an inductively defined set is defined by the rule instances, we are free to introduce metanotation i...