POSNER, Chief Judge.

A coal company asks us to set aside an award of benefits under the black-lung law, 30 U.S.C. § 901 et seq. Fitts, the claimant, was a miner for 19 years, and also a heavy smoker. The record before the administrative law judge consisted of reports by three physicians, Drs. Rao, Houser, and Tuteur, plus x-rays and other test results on which the physicians had relied. Rao and Houser concluded that Fitts had pneumoconiosis, Tuteur that he did not. Rao and Houser rested their conclusion, in part anyway, on x-rays that they interpreted as positive for pneumoconiosis. Later, more experienced radiologists ("B-readers") interpreted the x-rays as negative for pneumoconiosis. Despite this, the administrative law judge found that Fitts had the disease, because "While Dr. Tuteur's report is both adequately documented and well reasoned, his opinion is outweighed by the opinions of Drs. Rao and Houser, despite the subsequent negative x-ray interpretations." End of analysis.

This will not do. Administrative agencies when engaged in factfinding and law-applying are required to proceed in accordance with the elementary principles of rational truth-seeking; and the adjudication of black-lung cases is not exempt from this requirement. Peabody Coal Co. v. Old Republic Ins. Co., 972 F.2d 178, 182 (7th Cir. 1992). One such principle is that the tribunal shall not base its decision on a mechanical nose count of witnesses. Adkins v. Director, 958 F.2d 49, 52 (4th Cir. 1992); cf. Mullins Coal Co. v. Director, 484 U.S. 135, 149 n. 23, 108 S.Ct. 427, 435 n. 23, 98 L.Ed.2d 450 (1987). That procedure has been discredited for centuries. In modern conditions its only effect would be to multiply the number of witnesses. When as is normal in these cases the same x-ray is read by a number of different physicians, who note their interpretation but do not give reasons, the balance of opinions is entitled to some though not controlling weight. Old Ben Coal Co. v. Battram, 7 F.3d 1273, 1277-78 (7th Cir. 1993). It is difficult to see what alternative the administrative law judge would have. But we were astonished to find the Benefits Review Board of the Department of Labor, which reviewed the administrative law judge's award of benefits to Mr. Fitts, citing "numerical superiority" not of x-ray readings but of witnesses as a reason for affirmance. So far as appears from the passage we quoted from the administrative law judge's opinion, it was on the basis of that same principle of "numerical superiority" that he decided that Rao's and Houser's reports, despite their flaws, out-weighed Tuteur's, though the administrative law judge identified no flaw in Tuteur's report.

Of course the fact that Rao and Houser each relied on a questionable piece of evidence — an x-ray they though positive for pneumoconiosis but more experienced x-ray readers though negative — did not by itself invalidate their conclusions. When a witness relies for his conclusion on facts A, B, and C, and fact A is knocked out, it does not follow that his conclusion must change. It may be that his conclusion would be unchanged as long as two out of the three facts, or even just one of the three facts, were true. If this is plain there is no need to ask him to reconsider in light of the altered premise (not- A in place of A); but if it is not plain, then one must ask him to reconsider. See Thorn v. Itmann Coal Co., 3 F.3d 713, 718 (4th Cir. 1993).

No one reading either Rao's or Houser's report of their examination of Fitts would know what weight they had given the x-ray evidence. Rao's report, a printed form that he filled in, lists the results of the examination, and then, under the heading "diagnosis," states only, "(1) Severe C.O.P.D. [emphysema], (2) Pneumoconiosis." No one but a physician could tell from the earlier sections of the report how heavily Rao's conclusion rested on x-ray evidence. Houser's report is a little more expansive, stating that "Mr. Fitts has sufficient occupational exposure (approximately 20 years) and chest roentgenographic [x-ray] findings to support the diagnosis of coal workers' pneumoconiosis." Occupational exposure is not evidence of pneumoconiosis, however, but merely a reason to expect that evidence might be found. The only evidence that Houser mentions is x-ray evidence, over which the rereadings placed a cloud.

Now it is true that after being informed of the B-readers' rereading of his x-rays, Rao wrote the Department of Labor that "my initial impression of severe C.O.P.D. and Pneumoconiosis still stand[s] as is." There is no amplification, so it is unclear whether he merely disagreed with the B-readers' interpretation of the x-rays or though that other evidence of pneumoconiosis was compelling. And the administrative law judge did not mention or allude to the letter. But let us assume that it sufficed to rehabilitate Dr. Rao's finding. The record discloses no similar communication to and response from Dr. Houser. In their absence his report, which as we have said mentions no evidence of pneumoconiosis other than the x-ray evidence later read by more experienced readers as negative, not positive as he had thought, cannot be considered probative evidence that Fitts has pneumoconiosis. That leaves the witness score 1 to 1 — Rao versus Tuteur. Nothing in the administrative law judge's opinion offers a clue as to how to choose between the two physicians' opinions. In relying as he appears to have done on the principle of "numerical superiority," the administrative law judge overlooked the fact that even if the principle were valid, which it is not, it could not be applied in a case in which the witnesses were equally divided once one witness's opinion was discarded, as it must be in this case, for having a palpably inadequate basis in the facts.

The case is already more than six years old, and naturally we are reluctant to remand it. Mr. Fitts has very bad lungs, and there is certainly some, perhaps enough, evidence that their condition is due in part at least to what the statute classifies as "pneumoconiosis," though many members of the medical community would question the classification. Freeman United Coal Mining Co. v. Office of Workers' Compensation Programs, 957 F.2d 302, 303 (7th Cir. 1992). We hate to stretch the case out longer. But we are not authorized to affirm an administrative decision that is not reasoned. To base decision on which side produced more witnesses, and to include in the count of witnesses one whose opinion rested on a premise that was later discredited, is not a rational method of decision-making. The case must therefore be remanded to the Department of Labor for further proceedings.

The Director of the Office of Workers' Compensation Programs, a statutory respondent, has filed a brief that invites us to resolve an esoteric issue concerning proof of causation under the black-lung law, but admits that the resolution would not affect the outcome of this case. The Director is asking for an advisory opinion, useful in future cases but irrelevant to this one, and we are not authorized to issue such opinions. U.S. National Bank v. Independent Ins. Agents, ___ U.S. ___, ___, 113 S.Ct. 2173, 2178, 124 L.Ed.2d 402 (1993). So we must turn down the request.

VACATED AND REMANDED.