R. Hershel Schachter on Da'as Torah, in the name of R. Joseph B. Soloveitchik (Mi-P'ninei Ha-Rav, pp. 29-30):
In the time of the Shulhan Arukh, only one mourner would recite the mourner's kaddish. The Magen Avraham, in the name of latter decisors, explains all the rules of precedence and preferences on this matter [i.e. who would precede whom to recite kaddish]. It once happened that the Sha'agas Aryeh entered a synagogue and found two mourners debating who had the right to recite kaddish, and they asked the Sha'agas Aryeh to decide. When he heard the subject of their debate, the Sha'agas Aryeh suggested they make a lottery to decide who among them will win. There were in the synagogue at that time attendees who knew that there are rules in the posekim about precedence and they immediately asked him why he did not look in the Shulhan Arukh and rule accordingly. The Sha'agas Aryeh answered, "And how do you think the Magen Avraham reached his conclusions in this matter? We'll also make a lottery."However, it should be noted that neither R. Schachter nor R. Soloveitchik have said that wise people should not be consulted on difficult matters, and the greatest rabbis of the generation are usually among the wisest men.
Our teacher [R. Soloveitchik] would frequently tell this story in regard to matters that have no sources in halakhah. What relevance is there in making a halakhic decision on non-halakhic matters? (Compare Noda Bi-Yehudah YD 2:198 who wrote, "I am uncomfortable speaking about something that has no basis in the Talmud, because the Talmud is the source from which we draw theories and proofs. In a matter that has no origin, we cannot find proofs and foundational theories but only tentative theories." Similarly in Noda Bi-Yehudah YD 1:74 [sv. ve-al devar ha-nunin hafukhim], "For my whole life I have not liked to put forth anything that has no source in the Gemara and Posekim because about what is not explained in the Gemara we have no way to determine what is true.)
(Those interested in R. Hershel Schachter's legal philosophy, would do well to see the footnote in Mi-P'ninei Ha-Rav pp. 342-343 about the incorporation of science into halakhah regarding the ascertainment of fact. Compare his attitude towards DNA testing with that of many other posekim.)