Gemara

(Mishnah): Beis Din is liable only for Hora'ah that is a partial Akirah (uprooting of the Isur) and partial Kiyum (forbids like Torah law).

We learn from "and a matter was hidden", but the whole Mitzvah.

3b (Mishnah): If the Hora'ah totally uprooted a Mitzvah, e.g. Beis Din said that the Torah permits Nidah or Melachah on Shabbos or idolatry, they are exempt. If they permitted some Isurim, they are liable. E.g. they forbade Nidah, but exempted Bi'ah (relations) with a Shomeres Yom k'Neged Yom (a woman who saw blood during one of the 11 days of Zivah). Or, they forbade Melachah on Shabbos, but exempted one who transfers from Reshus ha'Yachid to Reshus ha'Rabim. Or, they forbade idolatry, but exempted one who bows.

4a (Shmuel): Beis Din is liable only if they permitted something that the Tzedukim (who reject our Oral tradition) permit. If even the Tzedukim forbid it, Beis Din is exempt. (Since it is explicit, no one should rely on the Hora'ah.)

Question #1 (Mishnah): (If Beis Din) forbade Nidah, but exempted Bi'ah with a Shomeres Yom k'Neged Yom (they are liable, even though this is explicit in the Torah - "v'Sofrah Lah")!

Answer: They exempted Ha'ara'ah (the first stage of Bi'ah) with her, or Bi'ah with one who saw blood at night.

Question #2 (Mishnah): If Beis Din forbade Melachah on Shabbos, but exempted one who transfers from one Reshus to another, they are liable.

Answer: They permitted only from Reshus ha'Rabim to Reshus ha'Yachid, or throwing or passing.

Question #3 (Mishnah): If they forbade idolatry, but exempted one who bows... (This is explicit - "Lo Sishtachaveh l'El Acher"!)

Answer: They forbade bowing only to idolatry normally served by bowing, or only if he spreads the hands and feet while bowing.

4b - Question (Rav Yosef): If they permitted plowing on Shabbos, what is the law? Since they acknowledge the other Isurim of Shabbos, this is like partial Akirah and partial Kiyum? Or, is this like total Akirah?

Answer #1 (Mishnah): If they ruled that one is liable for a Nidah, but exempts for a Shomeres Yom k'Neged Yom, Beis Din is liable (even though this is total Akirah of Shomeres Yom k'Neged Yom)!

Answer #2 (Mishnah): If they ruled that the Torah forbids Melachos on Shabbos, but exempts one who transfers from Reshus ha'Yachid to Reshus ha'Rabim (Beis Din is liable, even though this is total Akirah of Hotza'ah)!

Answer #3 (Mishnah): The Torah forbids idolatry, but one who bows is exempt. (They are liable, even though this us total Akirah of bowing!)

Rejections (Rav Yosef): We can answer all these like above.

Question (R. Zeira): If they ruled that Shabbos does not apply during Shemitah, is this is like uprooting part, or like uprooting the entire Mitzvah?

Answer (Ravina - Beraisa): If a prophet prophesizes to uproot a Mitzvah (he is a false prophet), he is killed;

R. Shimon says, he is exempt for partial Akirah. Regarding idolatry, even if he says to serve it today and abolish it tomorrow, he is killed.

Permitting Shabbos in Shemitah is like partial Akirah.

Rishonim

Rambam (Hilchos Shegagos 14:2): Beis Din is liable only if the Hora'ah is partial Akirah and partial Kiyum, in matters that are not explicit in the Torah. Then, Beis Din is liable and those who followed the Hora'ah are exempt. If they erred and ruled that one may bow to idolatry, or to transfer from Reshus to Reshus on Shabbos, or to have Bi'ah with a Shomeres Yom k'Neged Yom. This is like saying 'there is no Shabbos or idolatry or Nidah in the Torah', for they totally uprooted the Mitzvah. This is not a mistaken Hora'ah, rather, forgetting. Therefore, they are exempt from a Korban, and whoever relied on them is Chayav Korban. However, if they erred and ruled that one who transfers from Reshus to Reshus on Shabbos is liable, for it says "Al Yetzei Ish miMkomo", but one may throw or pass, or they uprooted an Av Melachah and said that it is not a Melachah, they are liable. Similarly, if they erred and said that one who prostrates to idolatry with spread hands and feet is liable, for it says "Lo Sishtachaveh l'El Acher", but one may bow on the ground without spreading hands and feet, they are liable. Similarly, if they erred and said that one who has relations with a Shomeres Yom k'Neged Yom who saw blood today is liable, for it says "Kol Yemei Zovah", but if she saw at night one may have Bi'ah with her.

Kesef Mishneh: The Gemara never resolved whether or not Beis Din is liable for permitting a Melachah. The Rambam should have ruled leniently (lest Beis Din bring Chulin b'Azarah, i.e. an invalid Korban)! Perhaps they stipulate 'if we are exempt, this is a Nedavah (voluntary Korban).' Alternatively, since the Gemara answered the question from three Mishnayos, and the questioner rejected the answer, we do not abandon what the Gemara held due to a Dichuy (mere rejection)

Question (Lechem Mishneh 13:4 and 14:2): In three places, the Rambam rules that Beis Din brings a bull when the Gemara did not resolve the Halachah. We cannot say that they stipulate, for the bull is a Chatas, and one cannot bring a Chatas Nedavah! In Hilchos Nezirus (10:4-6) the Rambam says that a stipulation helps for Olah, but for Chatas, the only solution is to bring Chatas ha'Of (birds), which may be brought due to Safek! Also the Kesef Mishneh's latter answer is difficult. The rejections are true, for we must say so to defend Shmuel! Kiryat Sefer says that even though the rejections were said like Dichuyim, they are really true. Permitting an entire Melachah is like totally permitting Shabbos, unless they fulfilled part of the Melachah, e.g. carrying from Reshus ha'Yachid to Reshus ha'Rabim. This is like I said. However, the Rambam (whom Kiryat Sefer comes to explain) says oppositely! Perhaps he he had a different text of the Rambam. Why didn't the Rambam bring the question about permitting Shabbos in Shemitah? The Rambam must hold that these questions are equivalent. Rav Yosef asked one of them, and R. Zeira asked the other. Therefore, when the Gemara resolved that the latter is partial Akirah, the same applies to the former. Therefore, it sufficed for the Rambam to rule about one of them.

Ri Korkus (DH v'Chasav): The Gemara never resolved whether or not Beis Din is liable for permitting a Melachah. Even so, the Rambam rules that they are liable, for the Gemara answered from our Mishnah that they are liable. Even though it rejected this, we do not abandon what the Gemara held due to a Dichuy. Also, the Gemara gave these answers only so it will not be something that Tzedukim admit to, so it will not be difficult for Shmuel. It seems that the question was only against Shmuel, but without Shmuel, we understand why the Mishnah is considered partial Heter. Also, the Gemara concluded that permitting Shabbos in Shemitah is like partial Akirah. All the more so, permitting part of Shabbos (one Melachah) and forbidding the rest is considered partial Akirah (for he does not totally permit a Shabbos)! One could say oppositely, that here he totally uprooted a Melachah. The Rambam holds that these are equal. Also, the Mishnah says that saying that there is no Nidah or Shabbos or idolatry in the Torah is called total Akirah (but not merely permitting one Melachah). Since the Gemara brought a proof fromth Seifa, it did not bother to bring a proof from the Reisha that can be rejected.

Keren Orah (4a DH umi'Divrei): Shmuel says that Beis Din must permit something that Tzedukim argue about. The Rambam holds that Shmuel explains the exclusion of a total Akirah. (See Perush ha'Mishnayos 1:3.) Beis Din is exempt because it is explicit in the Torah. It is a mistake, not a Hora'ah. Whatever is explicit is called total Akirah, even if the entire Mitzvah was not uprooted, e.g. Hotza'ah on Shabbos or bowing to idolatry, since the Torah explicitly forbids these. Partial Bitul is a ruling about something Chachamim expound, e.g. bowing with spread hands and feet, or throwing or passing. However, why did Rav Yosef ask about plowing? It is total Akirah only if it is explicit in the Torah! (Note: R. Yishmael holds that "be'Charish uva'Ketzir Tishbos" forbids plowing on Shabbos, and R. Akiva says that it discusses Shemitah. Rav Yosef connotes that his question does not depend on how to explain this verse.) The Rambam says that uprooting one Melachah is partial Akirah. He must say that Rav Yosef holds that plowing is explicit in the Torah, and asks whether only total Akirah exempts Beis Din. His question was not settled, but it is settled from Shmuel, that whatever is explicit is like total Akirah. However, the Rambam rules that the verse discusses Shemitah, and plowing is not explicit in the Torah. Beis Din is liable for Hora'ah about other Melachos, which are not explicit in the Torah.

Chazon Ish (Horayos 14:14 DH v'Efshar): Perhaps the Rambam holds that just like permitting bowing to idolatry without spreading the hands and feet is not called total Akirah, also permitting a Melachah.