Yevamot, Chapter One, Mishnah Four
Mishnah four teaches us that the first three mishnayoth of the tractate were all according the opinion of Beth Hillel. Whereas Beth Hillel ruled that a woman whose rival wife is forbidden to the yavam is exempt from yibbum (as is the forbidden wife), Beth Shammai ruled that she must have yibbum or halitzah. The mishnah then lists a few correlating disputes on this matter.
Finally, the mishnah sounds what is surely the most pluralistic note in the entire Mishnah: despite their many disagreements, Beth Hillel and Beth Shammai did not separate into two different sects.
I) Beth Shammai permits the rival wives to the surviving brothers, and Beth Hillel prohibits them.
II) If they perform the halitzah, Beth Shammai disqualifies them from marrying a priest, and Beth Hillel makes the eligible.
III) If they performed yibbum, Beth Shammai makes them eligible [to marry a priest], and Beth Hillel disqualifies them.
IV) Though these forbid and these permit, and these disqualify and these make eligible, Beth Shammai did not refrain from marrying women from [the families of] Beth Hillel, nor did Beth Hillel [refrain from marrying women] from [the families of] Beth Shammai.
A) [With regard to] purity and impurity, which these declare pure and the others declare impure, neither of them refrained from using the utensils of the others for the preparation of food that was ritually clean.
Section one: I have explained this section above in the introduction.
Section two: If the yavam should perform halitzah with the rival wife, according to Beth Shammai, this halitzah is valid and allows this woman to marry other men. Since a woman who has had halitzah is prohibited to a priest (just as a divorcee is forbidden to a priest), this woman is now prohibited to marry a priest. According to Beth Hillel, this was an unnecessary halitzah, for the woman could have married other men even without it. Since it was not true halitzah, she can still subsequently marry a priest.
Section three: If she has yibbum, full levirate marriage, Beth Shammai would consider this to be valid, as they stated in section one. Should this marriage end in the husbands death, she could marry a priest, as can all widows. However, according to Beth Hillel, this yibbum was actually forbidden. Therefore this woman can no longer marry a priest, since all women who have engaged in forbidden sex cannot subsequently marry priests (we will learn more about this law later in the tractate).
Section four: In Jewish law the child of many forbidden unions is considered to be a mamzer (we will also learn these laws later in the tractate). A mamzer cannot marry a regular Israelite (Jew). In the above dispute, if the rival wife did perform yibbum, the child would be a mamzer according to Beth Hillel. Since this was not a situation requiring yibbum, the woman was biblically forbidden to her dead husbands brother. If the rival wife married another man without yibbum or halitzah, the child of this subsequent marriage would be a mamzer according to Beth Shammai. This is because this woman married out while still bound to her yavam. This section teaches that although this dispute would have created children considered by some to be mamzers and others not to be, neither side refrained from marrying those from the other side.
This is a remarkable note of pluralism in the Mishnah, indeed it seems that this ideology is what kept the rabbis from forming splinter groups. Although Beth Shammai and Beth Hillel disagreed on many topics, they continued to live together and intermarry. Unlike other groups, specifically the Dead Sea sect, which left Jerusalem over halakhic disputes with their opponents, the halakhic disputes between the rabbis did not cause them to break into many little groups. The mishnah further teaches that despite the fact that things that Beth Hillel considered pure, Beth Shammai considered impure, and vice versa, they did not refrain from eating together and from using each others utensils. Again, if we look at the group that did split, the Dead Sea sect, two of the halakhic issues which they considered the most important were marital laws and purity laws (sacrificial laws were also important.) The authors of the Dead Sea scrolls are constantly accusing their opponents of not properly observing marriage laws or purity laws. Despite the fact that Beth Shammai and Beth Hillel also debated these subjects, they did not refrain from marrying each other and from eating together. In other words, they remained one society.