What did Igros Moshe 1.158 mean by the words "מדין שני בתי דינים בעיר אחת" ("from the law of two courts in one city")?
What is דין שני בתי דינים בעיר אחת?
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Rabbi Feinstein is addressing questions of local customs. Theoretically 400 years ago, you'd have all the Jews in Baghdad following the customs of Baghdad, and all the Jews of Krakow following the customs of Krakow. What happens today when a bunch of Krakow Jews and Baghdad Jews come together in New York City? What is the "local custom?"
The halacha is that if a city two independent rabbinic courts, one following Krakow practices and one following Baghdad practices, that there is room for both of them. Most notably, the two courts could have different ways of spelling the city's name in a Gett, and each are acceptable. If there is only one rabbinic establishment, however, then all the cities' residents are expected to abide by it in certain matters. An Ashkenazi who just happens to be passing through Djerba, Tunisia and needs a Gett would use one with the Hebrew spelling of "Djerba" that was developed by the Sephardic establishment there, even though it's very different than how Ashkenazim would transliterate it.
Rabbi Feinstein is noting that New York has multiple rabbinic establishments and thus there is no one set of "local customs" that everyone is expected to follow. 400 years ago, questions of mixed marriages were generally moot because "local city custom" trumped all: if you moved to Krakow, you followed Krakow customs! Today in New York, however, there are Krakow Jews and Baghdad Jews independently following their respective customs, so we have to figure out what to do when there's a marriage between the two communities. (Rabbi Feinstein's general guideline was to follow the customs of the husband's community, whether they are more lenient or less than those of the wife's community.)
The phrase originates in the Gemara, Yevamot 14a:
(דברים יד) לא תתגודדו לא תעשו אגודות אגודות אמר אביי כי אמרינן לא תתגודדו כגון שתי בתי דינים בעיר אחת הללו מורים כדברי ב"ש והללו מורים כדברי ב"ה אבל שתי בתי דינים בשתי עיירות לית לן בה אמר ליה רבא והא ב"ש וב"ה כשתי בתי דינים בעיר אחת דמי אלא אמר רבא כי אמרינן לא תתגודדו כגון ב"ד בעיר אחת פלג מורין כדברי ב"ש ופלג מורין כדברי ב"ה אבל שתי בתי דינין בעיר אחת לית לן בה
The Gemara is applying the verse "do not inflict gashes upon yourself" at the level of splintering communal practices. Rava comments that there were cities with parallel Hillel-ite and Shammai-ite rabbinic establishments and that they could co-exist. The problem is when there's a single rabbinic establishment but no coherent practice.