Many arguments that generate unending derivations have counterexamples that use only finitely many reference values. But, although the rules can be modified to uncover such counterexamples, this is not enough to insure decisiveness.
7.8.1. Finding finite structures
We can search for finite counterexamples by modifying rules to consider old terms along with new ones or to consider the possibility that new terms are co-aliases of old ones.
7.8.2. The failure of decisiveness
We cannot hope to find counterexamples in this way for all invalid arguments because the counterexamples to some invalid arguments always have infinite ranges.