1. At the top of Page 102(a) of Gemarah Yevamos on the fourth and fifth lines it states that a Ger judges his friend Ger but if his mother is an Israelite he can even judge an Israelite. Why would someone whose mother is a Jew need to convert? Does this Gemarah imply a difference of opinion as to whether a person who has a Jewish mothter and a non-Jewish father is really a Jew? In Talmudic times was there a doctrine that required a person to convert his "non-Jewish paternal half"? How else do you explain why the Talmud would mention that a Ger had a Jewish mother?
2. Can you tell me the Talmudic source (exact page number) which explains why Shavuos is two days outside Israel. How do we know that the second day of Shavuos did not begin to be observed well after the Talmud was compiled, perhaps in the time of the Geonim? Is Lo Ploog explicitly mentioned in the Gemarah in connection with Shavuos (whose timing was was never subject to doubt because it is based on the omer counting) or only in connection with the moon sighting holidays? The Rambam's reasoning is not complete (and in fact implies that Shavuos is doubt-based holiday, which of course it isn't) and does not cite a source for his assertion that the Rabbis established the second day of Shavuos.
Allan I. Kroll, Mt. Crested Butte, CO 81224
Due to recent abuse of our service, we generally no longer respond to questions from nonsubscribers without requesting identifying information.
However, we will briefly address your questions:
1: Although one who's mother is Jewish is a Jew and needs not convert, however he is considered a Ger with regard to Yichus, since he does not have a Jewish father. This is relevant to the laws of judges, as Rava states; a Ger may not judge another Jew unless his mother was a born Jewess. He may not perform Chalitzah unless both parents were Jewish. Rava is merely contrasting the law of judges with that of Chalitzah, as is clear from the context there. (There is no implication whatsoever that a Jew whose father was not Jewish must convert.)
2. The first source that comes to mind is Erchin 10a, which states that we say the full Hallel for 21 days out of the year, and it lists two days of Shavuos. As for why there are two days of Shavuos, the early authorities discuss this.