 (1) Counterexamples to the conclusion that are not also counterexamples to any of the premises (“global but not local counterexamples”): These establish that the conclusion does not really follow from the stated premises. They require us to improve the proof, to unearth the “hidden lemma” which the counterexample also refutes, so that it becomes a “local as well as global” counterexample—see (3), below.