Gemara

2a (Mishnah): All (may) slaughter, and the slaughter is Kosher, except for a Cheresh (deaf-mute), lunatic or minor, lest he slaughter improperly. If any of them slaughtered while others looked on, the slaughter is Kosher.

3b (Ravina): The Mishnah teaches 'all may slaughter', i.e. anyone who knows the laws, even if he is not established (that his hands are skilled, and that he does not get weary during the slaughter);

'Except for a Cheresh, lunatic or minor' teaches that if he slaughtered, the animal is forbidden, lest he paused during the slaughter, pressed (the knife, instead of cutting), or did Chaladah (slaughter when the knife is covered).

12b (Rava): 'Except for a Cheresh... lest he will slaughter improperly' teaches that l'Chatchilah, we do not let them slaughter Chulin.

Our Mishnah is Machshir if any of them slaughtered while others looked on. This is like R. Noson, who does not require intent for Shechitah of Chulin;

(Beraisa - R. Noson): If one threw a knife to insert it into a wall, and as it went it slaughtered normally, the Shechitah is valid;

Chachamim disqualify.

(Oshiya Ze'ira of Chavriya): The Halachah follows R. Noson.

86a (Mishnah): If a Cheresh, lunatic or child slaughtered while others looked on, the onlookers must cover the blood. If he slaughtered unsupervised, Kisuy need not be done.

Similarly, if a Cheresh, lunatic or child slaughtered while others looked on, one may not slaughter the mother or child of the animal that same day;

R. Meir says, if he slaughtered unsupervised, one may slaughter the mother or child the same day,

Chachamim forbid, but agree that if one did, he is not lashed.

Chachamim argue also in the Reisha. They are stringent in both cases (due to Safek).

(R. Yakov): R. Meir considers the slaughter of a Cheresh, Shoteh or Katan to be Nevelah.

This is because usually, they slaughter improperly.

R. Elazar said that Rebbi ruled like R. Meir. He did not mention that Rebbi also ruled like Chachamim, for that was earlier (and he retracted).

Sukah 42b (Beraisa): If a minor knows how to slaughter, one may eat from his slaughter.

(Rav Huna): This is if an adult supervised.

Gitin 22b (Mishnah): Anyone can write a Get, even a Cheresh (deaf-mute), lunatic or minor.

Question: Why is this valid? They lack Da'as (understanding)!

Answer #1 (Rav Huna): The case is, an adult supervises them.

Question (Rav Nachman): Would you say that adult supervision enable a Nochri to write a Get?! A Beraisa disqualifies it!

Answer: A Nochri is different. He writes for his own motive. (A Cheresh, lunatic or minor writes according to what the adult tells him.)

Yevamos 104b (Mishnah): If the Yavam or Yevamah was deaf, or he was a minor, the Chalitzah is Pasul.

Rishonim

Rosh (1:4): The Mishnah forbids a minor to slaughter l'Chatchilah with adult supervision. This is before the age of Chinuch (training in Mitzvos), i.e. when he can train his hands. If the minor is an expert and trained his hands, he may slaughter l'Chatchilah. In Sukah, we say that if a minor knows how to slaughter, one may eat from his slaughter. This connotes l'Chatchilah.

Beis Yosef (YD 1 DH veha'Rosh): At first he says that Chinuch is when he can train his hands, and this suffices to slaughter l'Chatchilah with adult supervision. Later, he says 'if he is an expert and trained...', which connotes that we require both. The Tur answers that indeed, if he trained his hands he may slaughter l'Chatchilah with adult supervision. Later, he teaches that if he is also an expert, l'Chatchilah he needs adult supervision, but b'Di'eved, if there was not, the slaughter is Kosher.

Rebuttal (Taz 14): The Tur allows l'Chatchilah only with both qualifications! It seems that to answer the contradiction, the Rosh said that in Chulin the child is an expert but did not train his hands, therefore we do not give to him l'Chatchilah. In Sukah, he is an expert and trained his hands.

Hagahos Ashri: Even if the minor is an expert and trained his hands, if no adult supervised, the slaughter is Pasul.

Tosfos (Sukah 42b DH veha'Yode'a): Rashi says that the minor trained his hands to slaughter, but he does not know the Halachos, therefore an adult must supervise that he does not delay or press. Rashi connotes that if he did not train his hands to slaughter, even if an adult saw the entire slaughter that he did not delay or press, one may not eat it. It seems that Semag and Sefer ha'Terumos agree.

Ran (Sukah 20b DH Yode'a): Why must the minor know how to slaughter? The Mishnah permits when others oversee him! We must say that that is b'Di'eved. If he trained his hands, he may slaughter l'Chatchilah.

Rashba (1:26, brought in Beis Yosef (OC 460 DH Ein): Matzah must be guarded, therefore one is not Yotzei with Matzah kneaded by a Nochri, Cheresh, lunatic or child. Adult supervision does not help for a Nochri, for he acts for himself. We find that a Cheresh, lunatic or child can write a Get only if he is supervised. Supervision does not help for Shechitah, according to Chachamim who require intent. Tosfos (12b DH Man) says that others see, but do not teach him and admonish him to intend, so it is unlike 'Omed Al Gabav' that helps for a Get. However, this does not distinguish a lunatic from a Cheresh or child. Regarding Chalitzah, we distinguish them. Chalitzah of a deaf or minor Yavam is Pasul and forbids Yibum. We do not say so about Chalitzah of a lunatic. Tosfos says that a lunatic has no Da'as (understanding) at all, and supervision does not help at all if there is no indication (of intent) in the matter he does, e.g. Shechitah and Chalitzah. Regarding a Get, they tell him to write for Ploni and Plonis, and he does. He musters up intent and writes Lishman (for their sake). According to this, with supervision, a Cheresh or child, who has a little Da'as, can knead for Matzah, and we are Yotzei with it. One is not Yotzei with a lunatic's Matzah.

Poskim

Shulchan Aruch (OC 460:1): We do not knead Matzos for the Mitzvah through a Nochri, Cheresh, lunatic or child.

Rashba (ibid.): R. Yonah says that for anything that can be done through a Shali'ach, such as Shechitah or writing a Get, intent of a supervisor (David) helps like intent of the one doing it (Levi). It is as if Levi is David's Shali'ach. Therefore, a lunatic can slaughter or write a Get if a Pike'ach (healthy person) supervises. Chalitzah cannot be done through a Shali'ach, therefore a Cheresh, lunatic or child cannot do Chalitzah, even if a Pike'ach supervises. According to this, the same applies to kneading l'Shem Matzas Mitzvah. However, it seems that it depends only on whether or not there is an indication that the act is done Lishmah. Writing a Get with the proper names is an indication, so even a lunatic can do so if a Pike'ach supervises. There is no indication in Shechitah or kneading, so supervision cannot help. There is a small indication in Chalitzah, i.e. the supervisors instruct them, therefore it helps for a Cheresh or child, but not for a lunatic, who has no Da'as.

Shulchan Aruch (YD 1:5): L'Chatchilah, we do not give animals to slaughter to a Cheresh, lunatic or a minor who did not train his hands to slaughter, even if an adult supervises. If they slaughtered with adult supervision, it is Kosher. If a minor trained his hands to slaughter, l'Chatchilah we give to him to slaughter with adult supervision and one may eat from it.

Shach (25): The Beis Yosef is always Machshir with adult supervision, and we give to him l'Chatchilah if he is an expert or trained his hands. We always forbid b'Di'eved without supervision, for he is not believed. The Yam Shel Shlomo (1:3) and Drishah (8) permit l'Chatchilah only with both qualifications, and forbid b'Di'eved without either qualification, even if there was supervision.

Rema: If he slaughtered by himself, it is Pasul even if he knows the Halachos of Shechitah. He is called a minor until 13 years.

Bedek ha'Bayis: We may rely on minors only regarding Isurim mid'Rabanan, like it says in Pesachim 4b.

Gra (27): Ravina explained that the Mishnah says 'all may slaughter' - anyone who knows the laws, 'except for a Cheresh...', i.e. even if he knows the laws. Rava derived that l'Chatchilah, we do not let a Cheresh... slaughter Chulin. If he does not know the Halachos, the same applies to adults!

Taz (15): Why does the Tur say that it is Kosher? We do not believe a minor about Tevilas Kelim! It seems that that is when we know only through the minor. Here, we see that it was slaughtered, just we do not know whether it was a proper Shechitah. For this we rely on a trained expert minor. Mishmeres ha'Bayis says that we rely on sharp minors to buy meat and wine. This is not testimony, rather, a Chazakah that they will buy Heter. Other Poskim do not bring the Tur's Heter.