1. If all the physicians are either hematologists or neurologists, then there are no cardiologists. But Dr. Frank is a cardiologist. Therefore, some physicians are not neurologists. (P, H, N, C)

2. Either Dr. Adams is an internist or all the pathologists are internists. But it is not the case that there are any internists. Therefore, Dr. Adams is not a pathologist. (I, P)

3. If some surgeons are allergists, then some psychiatrists are radiologists. But no psychiatrists are radiologists. Therefore, no surgeons are allergists. (S, A, P, R)

4. Either some general practitioners are pediatricians or some surgeons are endocrinologists. But it is not the case that there are any endocrinologists. Therefore, there are some pediatricians. (G, P, S, E)

5. All physicians who did not attend medical school are incompetent. It is not the case, however, that some physicians are incompetent. Therefore, all physicians have attended medical school. (P, A, I)

Translate the following arguments into symbolic form. Then use the quantifier negation rule and the eighteen rules of inference to derive the conclusion of each. Do not use either conditional proof or indirect proof.