HOW DO WE EXPLAIN 'OTHERS'?
Suggestion: Others hold that slaughter is only at the end (the moment that the majority of the second Siman is cut (see note in Appendix) - everything before this is only preparation - therefore, only one intent takes effect, the first);
This is like Rava, who says that the same argument about two different intentions in slaughter (R. Meir follows the first intent, R. Yosi follows both) applies to one who makes Temurah (see note in Appendix) 'l'Chatzos' (half should be Temuras Olah, half should be Temuras Shelamim - slaughter and Temurah are both instantaneous);
Therefore, only the first intent takes effect, be it Mulim or Arelim ('Others' is R. Meir after he tried to depose R. Shimon ben Gamliel - Horayos 13B. This implicitly challenges Abaye, who holds that the entire cutting from the beginning to the end is considered slaughter - presumably, Chachamim [who Machshir in either case - Mishnah 61A] hold like R. Yosi, but agree that slaughter is only at the end!)
Rejection (and Answer #1 - Rabah): No - Others hold that slaughter is from the beginning to the end - the case is, he intended for both of them, but did not finish saying the second intent until after slaughter;
R. Meir does not require equality of [words uttered by his] mouth and heart (intent) - therefore, the first intent [which he said before slaughter] takes effect by itself, even though he did not want this; (Tosfos - this shows that Pigul [and all Pesulim of intent] apply only if one verbalized the intent - otherwise it would not matter when he said it, only when he intended.)
Chachamim require equality of mouth and heart - therefore, his words do not take effect.
Objection: R. Meir does require equality of mouth and heart!
(Mishnah): If one intended to say '[this is] Terumah' and mistakenly said 'Ma'aser', or vice-versa, or [intended to vow or swear] 'I will not enter this house' or 'I will not benefit from Ploni', and he mistakenly said another house or person, his words do not take effect, unless there is equality of mouth and heart. (An anonymous Mishnah is assumed to be R. Meir.)
Answer #2 (Abaye): In the Reisha, he said that he slaughters the first Siman for Mulim and the second even for Arelim - the second Siman is for both (joint intent does not Posel);
In the Seifa, he said that he slaughters the first Siman [solely] for Arelim and the second for Mulim.
This is as R. Meir holds elsewhere, that improper intent for one of two Matirim (see note in Appendix) makes Pigul [like intent in all the Matirim - likewise, intent in half the slaughter is Posel like intent in the entire slaughter];
Chachamim hold like they taught elsewhere, that intent in one of two Matirim does not make Pigul.
SLAUGHTERING 'ON' CHAMETZ
(Mishnah): If one slaughters Korban Pesach 'on' (this will be explained) Chametz, he transgresses a Lav ("Lo Sishchat Al Chametz Dam Zivchi");
R. Yehudah says, this also applies to the [afternoon] Tamid [on Erev Pesach].
R. Shimon says, if one slaughtered Pesach [on Chametz] on the 14th Lishmah, he is liable; if he slaughtered it Lo Lishmah, he is exempt [for the slaughter is Pasul - R. Shimon holds that a Pasul slaughter is not called slaughter];
One is exempt for any other Korban, whether Lishmah or Lo Lishmah [even if it is Kosher - this will be explained];
During Pesach, if one slaughtered Pesach [on Chametz] Lishmah, [it is Pasul, so] he is exempt; if he slaughtered it Lo Lishmah, [it is Kosher, so] he is liable;
One is liable for any other Korban, whether Lishmah or Lo Lishmah (it is Kosher), except for Chatas Lo Lishmah, for which he is exempt (it is Pasul).
(Gemara - Reish Lakish): He is not liable unless there was Chametz in the Azarah belonging to the slaughterer, the Zorek (the one who will do Zerikah), or an owner of the Korban;
(R. Yochanan): He is liable even if the Chametz was not in the Azarah.
Question: What do they argue about?
Answer #1: Reish Lakish holds that 'Al' (in "Lo Sishchat Al Chametz") connotes nearby, R. Yochanan holds that it does not.
Objection: They argued about this elsewhere (they would not need to argue about it in two places)!
(Mishnah): (Todah must be slaughtered "Al Chalos.") If Lachmei Todah were outside the wall when the Todah was slaughtered inside, the bread is not Mekudash;
Question: What does it mean 'outside the wall'?
Answer #1 (R. Yochanan): It means, outside the wall of Beis Pagi (Rashi - outside of Yerushalayim; Rambam - outside of Har ha'Bayis).
Within Beis Pagi, even outside of the Azarah, it becomes Kodesh - "Al" does not connote nearby.
Answer #2 (Reish Lakish): It was outside the wall of the Azarah.
This is because he holds that 'Al' connotes nearby.
Answer #2: They argue about Hasra'as Safek (warning someone that he might transgress - Reish Lakish exempts if the Chametz is not there, for otherwise we are not sure if it is intact).
Objection: They argued also about this elsewhere!
(R. Yochanan and Reish Lakish): If Reuven swore 'I will eat this loaf today' and he did not eat it, he is not lashed;
(R. Yochanan): He is not lashed because it is a Lav she'Ein Bo Ma'aseh (one transgresses through inaction) - one is not lashed for such a Lav;
He does not exempt on account of Hasra'as Safek (perhaps he will not transgress, i.e. he will eat the loaf later) - he holds that it is valid warning.
(Reish Lakish): He is not lashed because Hasra'as Safek is invalid Hasra'ah;
He does not exempt on account of Lav she'Ein Bo Ma'aseh -- he holds that one is lashed for such a Lav.
Defense of Answer #1: Really, they argue about whether or not 'Al' connotes nearby - their argument must be taught in both cases:
Had they argued only about Chametz, one might have thought that there R. Yochanan does not require it to be nearby, for wherever it is the owner transgresses [Bal Yera'eh and Bal Yimatzei];
But he would agree that Lachmei Todah must be inside to become Mekudash, for 'Al' connotes nearby, just like we find that a Keli Shares is Mekadesh only what is inside it.
Had they argued only about Lachmei Todah, one might have thought that there Reish Lakish requires it to be nearby - but he would agree regarding [slaughtering on] Chametz, for wherever it is the owner transgresses.
Question (Rav Oshaya): If the slaughterer does not have Chametz but one of the Benei Chaburah does, what is the law?
Answer (R. Ami): It does not say 'Lo Sishchat Al Chamtzecha (your Chametz)', rather, "Lo Sishchat Al Chametz" (any Chametz)!
Question (Rav Oshaya): If so, he should transgress if anyone in the world has Chametz [in the Azarah]!
Answer (R. Ami): It says "Lo Sishchat...v'Lo Yalin" - do not slaughter on Chametz of anyone who is commanded not to leave over from this Korban (the Benei Chaburah).
(Rav Papa): Therefore, also the Kohen who is Maktir (burns on the Mizbe'ach) the Chelev transgresses [if he has Chametz then], since he is commanded not to leave the Chelev overnight.
Support (Beraisa #1): One who slaughters Pesach on Chametz transgresses a Lav;
This is if the slaughterer, the Zorek, or a member of the Chaburah had Chametz; we are not concerned for Chametz of other people.
The slaughterer, the Zorek, and the Maktir transgress.
One who is Molek a bird (see note in Appendix) on the 14th [on Chametz] does not transgress anything. (We will establish this to be like R. Shimon, who exempts all Korbanos at this time except for Pesach - the Beraisa discusses a typical case, for birds of a Mechusar Kipurim may be offered at the time of Pesach.)
Contradiction (Beraisa #2): If one slaughters Pesach on Chametz, he transgresses a Lav;
R. Yehudah says, this also applies to the Tamid;
Chachamim: It only applies to the Pesach.
This is when the slaughterer, the Zorek, or a member of the Chaburah had Chametz; we are not concerned for Chametz of other people.
One is liable for slaughter, Zerikah, Melikah or Haza'ah [sprinkling blood of a bird on the Mizbe'ach];
One who takes a Kometz (a handful, which is burned on the Mizbe'ach) of a Minchah or is Maktir Eimurim does not transgress a Lav;
The Beraisos contradict each other regarding Melikah and Haktaras Chelev!