Gemara

(Beraisa #1): If the blood became Tamei and Zerikah was done, if it was b'Shogeg, it is Meratzeh. If it was b'Mezid, Lo Hurtzeh.

This applies to a Korban Yachid. A Korban Tzibur is Meratzeh even b'Mezid.

Contradiction (Beraisa #2): The Tzitz is Meratzeh for blood, meat and Chelev that became Tamei, whether b'Shogeg or b'Mezid or b'Ones, for a Korban Yachid or a Korban Tzibur.

Resolution #1 (Rav Yosef): Beraisa #1 is like Chachamim. Beraisa #2 is R. Yosi:

(Beraisa): One may not separate Tamei produce to be Terumah on (to exempt) Tahor produce. If this was done b'Shogeg, it becomes Terumah. If it was done b'Mezid, it is not Terumah;

R. Yosi says, in either case, it is Terumah.

Objection: Granted, R. Yosi does not fine one who separated Tamei Terumah, but he holds (in a Beraisa) that the Tzitz is not Meratzeh for eating!

Resolution #2 (Rav Chisda): Beraisa #1 is Chachamim, and Beraisa #2 is R. Eliezer.

(Beraisa - R. Eliezer): Whether it was done b'Shogeg or b'Mezid, it becomes Terumah.

Resolution #3 (Ravina): Beraisa #2 discusses how it became Tamei (the Tzitz is Meratzeh even for Mezid). Beraisa #1 discusses the intent during Zerikah.

Resolution #4 (Rav Shila): Beraisa #1 discusses how it became Tamei, Beraisa #2 discusses the intent during Zerikah.

Beitzah 4b - Rav Masnah: If wood fell from a tree into an oven on Yom Tov, one may add to it a larger amount of Heter wood and light all of it.

The Muktzeh wood is Batel in the majority.

One may not be Mevatel a Torah Isur l'Chatchilah, but one may Mevatel an Isur mid'Rabanan l'Chatchilah.

Gitin 54b (Beraisa - R. Meir and R. Yehudah): (There was a mixture of forbidden Parech nuts, i.e. an esteemed species, with permitted nuts. Whole Parech nuts are never Batel in a mixture.) The nuts fell and broke. Whether this was Shogeg or Mezid, the mixture is still forbidden;

R. Yosi and R. Shimon say, if they fell b'Shogeg, the nuts can become Batel (in the proper amount of Heter). If they fell b'Mezid, they are not Batel.

Zevachim 74a (Rav Nachman): If a ring of idolatry was mixed with 100 rings and one of them fell into the sea, all the rest are permitted. We assume that the forbidden one fell.

74b (Reish Lakish): If a (sealed) barrel of Terumah became mixed with 100 Chulin barrels, and one fell into the sea, the rest are permitted.

(R. Elazar): If a barrel of Terumah became mixed with 100 barrels, we open one of them, remove Kedai Demai (the proper amount of Terumah) and one may drink the rest of the barrel.

Objection (Rav Nachman): This cannot be! (If so, closed barrels are not Mekadesh.)

Correction: Rather, if one of them was opened, we remove Kedai Demai, and one may drink the rest of it.

Bechoros 36 (R. Ila'a): If a Kohen says that an animal is a Bechor, and we would not have known this otherwise, he is believed to say that the Mum came by itself.

Question: We already learn this from a Mishnah!

(Mishnah): If a woman said, "I was married, then I was divorced" (and we had not known that she was ever married), she is believed, for without her we would not know of any Isur.

Answer: One might have thought that there she is believed, for she could have kept quiet. A Kohen must say that it is a Bechor, so a Chacham will (inspect the Mum and) permit it, for he does not want to eat Kodshim outside the Mikdash.

Rishonim

Rambam (Hilchos Ma'achalos Asuros 15:25): One may not be Mevatel an Isur Torah l'Chatchilah. If one was Mevatel it is permitted. However, Chachamim fined him and forbade everything.

Mishneh l'Melech: Tosfos (below) says that if one thought that it is permitted, this is like Shogeg. The Rambam in Hilchos Terumos connotes that otherwise. This requires investigation.

Rambam (Hilchos Terumos 5:8): One may not separate Tamei Peros to be Terumah on Tahor Peros. If one did so b'Shogeg, it is Terumah. If he knew that it is Tamei, but he thought that this is permitted, this is like Mezid.

Tosfos (25b DH b'Shogeg): B'Shogeg his Terumah is valid only in any case that it had Sha'as ha'Kosher, i.e. it was once Tahor when detached, but not if a Tamei person picked it while it was still wet (and able to receive Tum'ah). Then, it does not become Terumah (Pesachim 33a).

Rosh (Teshuvah 20:17): One day, many animals were slaughtered. Some were Treifos. The lungs of the Kosher animals were hanging on a pole, and it was found that one of them was Tereifah. Perhaps the one who checked erred, and one of the animals declared Kosher is really Tereifah. Or, perhaps he did not err, just the lung of a Tereifah became mixed with the Kosher lungs. A whole animal is not Batel. However, if one of them was already eaten, all are permitted, like it says in Zevachim about a ring or barrel that fell into Yam ha'Melech. The Ri said that this is only if it fell, but if he cast one in, even if he was Shogeg, we fine, due to Mezid. Here is different. If an animal was eaten before the Safek was known, we permit all of them. The Chazakah was that everything is permitted. A fine applies only after it is known that there is a mixture. Perhaps he will never find out that there is a mixture! Also, if one of the animals was cut into small pieces that are not Re'uyah Lehiskaved (a nice portion fitting to honor someone), before or after the Safek was known, all are permitted. This is like it says in Zevachim. When all the barrels are sealed, all are forbidden. If one was opened, we assume that it was the Terumah.

Poskim

Shulchan Aruch (YD 99:5): One may not be Mevatel an Isur l'Chatchilah.

Shach (7): The Rashba and Ran say that the Isur to be Mevatel an Isur is only mid'Rabanan. The Ra'avad holds that the Torah forbids this. The Ra'avad agrees about b'Di'eved.

R. Akiva Eiger: This affects what they hold about b'Di'eved. The Ra'avad holds that we fine Shogeg due to Mezid for an Isur Torah. It is permitted only if the Isur was already Batel (mid'Oraisa) in a majority of Heter before he added.

Pischei Teshuvah (3): The Noda bi'Yehudah (2 YD 45) says that the Isur to be Mevatel taste is mid'Rabanan, but for a dry Isur it is mid'Oraisa.

Shulchan Aruch (ibid.): Even if Isur fell into Heter without a Shi'ur to be Mevatel it, one may not add Heter in order to be Mevatel it. If he transgressed and was Mevatel it, or added, if he was b'Shogeg, it is permitted. If he was Mezid, it is forbidden to the Mevatel, if it is his, and to someone for whom he was Mevatel.

Beis Yosef (DH Kasav b'Sefer): Semag says that the Chacham must investigate whether the water was put in the pot after the Isur was put in. Semak agrees. The Tur says that even if he added afterwards, if he did not intend to be Mevatel the Isur, it joins (to be Mevatel). We must say that Semag is concerned lest there was not initially 60 times the amount of Isur in the pot, and the entire pot became forbidden like a Neveilah, and when Heter is added, we require 60 times everything in the pot. If one would not investigate, surely he would permit, for now there is 60 times the amount of Isur. If he finds that water was added after the Isur fell in, we need 60 times everything in the pot. If there is this amount, Semag agrees that it is permitted if it was added b'Shogeg. We hold that any Isur to which Heter was added b'Shogeg is permitted (if there is now a Shi'ur for Bitul). Semag and Semak hold that for all Isurim, Chatichah Na'asis Neveilah (a piece that became Asur due to absorptions forbids 60 times its volume, as if it were pure Isur). Many Poskim say that it applies only to meat and milk. Therefore, the Tur's opinion is primary.

Beis Yosef (DH Aval): If one intended to add, even Min b'Mino, it is not Batel. Beitzah 4b connotes that one may not add. The Tur permits if one added b'Shogeg. He learns from Gitin 54b; the Halachah follows R. Yosi and R. Shimon. In Siman 110, if something was not Batel due to its importance, and one of the mixture fell into Yam ha'Melech, the Tur permits the rest. The Ri says that this is only if it fell, but if he cast one in, even if he was Shogeg, it is forbidden, due to Mezid. The Rosh agrees. It seems that they rule like R. Meir and R. Yehudah, who forbid Shogeg due to Mezid. Why do they rule like them against R. Yosi and R. Shimon? Also here the Tur permits Shogeg, but in Siman 110, he rules like the Ri! Also, why did he challenge Semag? Semag forbids Shogeg due to Mezid, like the Ri! Perhaps we can distinguish Mevatel Isur from casting into the sea. We fine Shogeg only regarding the latter.

Rebuttal (Taz 8): The Tur did not learn from Gitin. It is an explicit Yerushalmi. (Terumos 5:3). R. Yochanan rules like R. Yosi and R. Shimon. When the Heter depends on increase (of Heter) or (nuts) breaking, we see now a situation of Bitul. Had the person been quiet, we would have permitted. Perhaps it was like this when the Isur fell in! We believe that he was Shogeg, for he could have kept quiet. In Siman 110, we see a situation of Isur, but he says that one of them fell to the sea. Had he been quiet, we would have forbidden. We do not believe him to permit. Bechoros 36a distinguishes like this.

Taz (7): In YD 84:13, we say that if he intended for something else, e.g. to fix the honey, it is permitted. This is if he had no other way to fix it. If he intended to be Mevatel, but he thought that this is permitted, it is permitted. One who erred about the Halachah is considered Shogeg.

Taz (9): The Tur connotes that it depends on whether he intended to be Mevatel the Isur. If he did, even if he thought that it is permitted, he is called Mezid. The Maharshal supports this. However, this is not so. If so, what was the Tur's question against Semag? We must investigate if he intended to increase, but thought that this is permitted! Rather, also this is called Shogeg. If one forgot a Melachah, but remembered that it is Shabbos, he is called Shogeg (Shabbos 67b). Shabbos is not different, for we bring a proof from it to other Isurim (Gitin 53-54). If a woman was Mezanah but she thought that it is permitted, Maharik (167) discusses whether this is called Mezid, for it says "she trespassed against her husband", and not 'she trespassed against Hash-m.' In other cases, one who erred about the Halachah is considered Shogeg. Therefore, I say that the Tur means that he intended to increase, and he knew that this is forbidden. Tosfos (Bechoros 23a DH Savar) says that even according to the opinion that fines Shogeg due to Mezid, it is considered Shogeg if he thought that one may be Mevatel.

Shach (8): The Shulchan Aruch teaches that not only is it forbidden to mix Isur into Heter to be Mevatel it. Rather, even if Isur fell into Heter, one may not add in order to be Mevatel it.

Sifsei Da'as (12 DH v'Im): If one was intentionally Mevatel but he thought that this is permitted, the Taz permits, and the Pri Chodosh considers this like a regular Shogeg. Some say that the Rambam in Hilchos Terumos retracted from Perush ha'Mishnayos, in which he ruled like the first Tana, and here he rules like R. Yehudah. I say that he did not retract. In the Mishnah, he explained R. Yehudah's words 'if he knew and forgot, it is not Terumah.' The Rambam explains that he knew that it is Tevel, and then forgot this. Regarding this, the Halachah does not follow R. Yehudah. If he knew that it is Tamei and forgot the law, the Rambam learns from the Yerushalmi that this is close to Mezid. Based on Tosfos in Menachos, we can say that the decree about separating Tamei Terumah is not due to the Kohen's loss, rather, lest one separate Terumah that had no Sha'as ha'Kosher. Therefore, we decreed about Shogeg close to Mezid, but we cannot learn to other Isurim. Also, we are more stringent about Terumah, for which there is Misah (b'Yedei Shamayim for a Zar or Tamei who eats it).

Gra (6): The Yerushalmi says that for any Isur, if one added b'Shogeg it is Batel. If one added b'Mezid it is forbidden. We cannot learn from Terumah.

Gra (7): The Halachah follows R. Yosi against his colleagues.