Gemara

(Mishnah): Everyone must fulfill Re'iyah, except for a Cheresh, lunatic or child...

2b (Ravina or Rava, and a Beraisa): One who can speak but not hear or vice-versa is exempt from Re'iyah, but he is obligated in Simchah. One who cannot hear or speak is exempt even from Simchah, and from all Mitzvos of the Torah.

Gitin 59a (Mishnah): A Cheresh gestures or reacts to gestures (to make transactions);

Ben Beseirah says, he is Kofetz or responds to Kefitzah for Metaltelim.

Gitin 71a (Rav Kahana): If a Cheresh can communicate by writing, we write and give a Get to his wife if he requests this.

Question (Rav Yosef): Why must Rav Kahana teach this? We know this from our Mishnah!

(Mishnah): If a man became unable to speak... and he nodded to show that h wants to divorce his wife, we test him.... If he answers properly, we write and give the Get.

Answer (R. Zeira): The Mishnah discusses a mute. He is not like a (total) Cheresh:

(Beraisa): One who can speak but not hear is a (partial) Cheresh. One who can hear but not speak is a mute. Both of them are like Pikchim (sound-minded people) in every respect.

Question (Beraisa): Regarding Metaltelim we rely on gestures of a Cheresh and what he writes, but not for Gitin.

Answer: Tana'im argue about whether or not he can divorce:

(Seifa - R. Shimon ben Gamliel): This applies only to one who was deaf from birth, but if a hearing man became deaf, he writes (a Get) and witnesses sign it.

71b (R. Yochanan): R. Shimon ben Gamliel argues with Chachamim.

72a (Beraisa): If a scribe wrote a Get Lishmah... it is Batel unless the scribe and witnesses heard the husband's voice command them.

'The husband's voice' teaches unlike Rav Kahana.

Yevamos 113b - Question (Rav Ashi): What is the reason of R. Elazar (who considers Terumah separated by a Cheresh to be Safek Terumah, because he is a Safek Pike'ach)?

Does he hold that a Cheresh always has the same weak Da'as (understanding), and he is unsure whether or not his Da'as is clear?

Or, is he sure that his Da'as is weak and (normally) unclear, but at times it is clear?

This determines whether he may divorce his wife. If he is always the same, his Get is as valid as his Kidushin. If he alternatives, he can Mekadesh but he cannot divorce. (Perhaps his Da'as was clear at the time of Kidushin, but unclear when he divorced!)

Rishonim

Rif and Rosh (Gitin 33a and 7:19): A Get is valid only if the scribe and witnesses heard the husband's voice command them. This teaches unlike Rav Kahana, who said that if a Cheresh can communicate by writing, we write and give a Get to his wife.

Gra (EH 120:15): Rav Kahana holds like R. Shimon ben Gamliel. The Beraisa unlike Rav Kahana shows that Chachamim argue with R. Shimon, like R. Yochanan taught.

Rambam (Hilchos Ishus 2:26): Whenever we discuss a Cheresh, this is one who cannot hear or speak. If he can do either of these he is like a regular person.

Shenos Eliyahu (Terumos 1:2 DH Cheresh she'Dibru): The Yerushalmi says that the Kelal (Terumos 1:2) that 'Cheresh' always refers to a deaf-mute is not always true. The Mishnah (Chagigah 2a) exempts a Cheresh from Re'iyah. We exempt from "Lema'an Yishme'u ul'Ma'an Yilmedu" even one who cannot hear or speak (we read Yilmedu like 'Yelamedu'). A Mishnah teaches that Chalitzah of a Cheresh or Chereshes is Pasul, i.e. because they cannot say the verses. The Bavli accepts the Kelal. It says that the Mishnah in Chagigah exempts from Re'iyah and Simchah only a Cheresh who is like a child and lunatic, i.e. without Da'as. A Cheresh who can hear or speak is like a Pike'ach in every way. He is exempt from Re'iyah, but must fulfill Simchah. The Kelal applies only when Cheresh, lunatic and child are taught together, therefore the Mishnah in Yevamos does not refute the Kelal. However, a Mishnah in Megilah disqualifies a Cheresh, lunatic or child for Kri'as ha'Megilah. There, it cannot mean a deaf-mute (who cannot read aloud)! The Yerushalmi tried to prove from there that R. Yosi requires that one hear himself say the words in all Mitzvos, not only in Kri'as Shma. It rejected this by saying that there, Cheresh (refers to a deaf-mute. It does not teach anything, it) was merely included in the Mishnah along with lunatic and child.

Rambam (Hilchos Gerushin 2:16): If a man became mute but his mind is sound, and he nodded his head when asked if we should write a Get for his wife, we test him well. If we see that he is not crazed, we divorce her. Similarly, If he wrote 'write and give a Get to my wife', if his mind is settled we do so. One who became mute is not like a Cheresh.

Kesef Mishneh: A Beraisa teaches unlike Rav Kahana, who said that if a Cheresh can communicate by writing, we write and give a Get to his wife. The Gemara (71a) connotes that if he can hear or speak, he can authorize a Get through writing. It seems that the Rambam agrees that we rely on gestures and writing of anyone except for a Cheresh, who is like a lunatic. The Rambam does not discuss one who can speak but not hear. It seems that since he is not a total Cheresh, surely he is like a mute.

Rambam (17): If a Pike'ach married and became a Cheresh, and all the more so if he became a lunatic, he cannot divorce until he gets healthy. We do not rely on his gestures or writing, even if his mind is sound and settled.

Rambam (Hilchos Edus 9:11): A Cheresh is like a lunatic. He is exempt from all Mitzvos of the Torah. Even if a Cheresh can speak but not hear or vice-versa, he cannot testify, even if he sees fine and has a sound mind. Testimony in Beis Din must be oral, or by one who could testify orally and hear the judges' warning.

Radvaz: The Rambam means that the Cheresh whom Chachamim always discuss, who cannot hear or talk, is exempt from all Mitzvos like a lunatic. A Drashah teaches that no Cheresh can testify, even if he can speak or hear and is obligated in Mitzvos.

Rambam (Hilchos Mechirah 29:2): One who cannot hear or speak, or if he can speak but cannot hear at all, can buy and sell Metaltelim through gestures, but not land. Even for Metaltelim we must test him thoroughly before validating his transactions.

Rambam (3): If one can hear but cannot speak, or if a person became mute, his purchase, sale and gift of land or Metaltelim is valid. We must test him like we test for Gitin.

Question (Magid Mishnah 2): The Mishnah (Gitin 59a) discusses a Cheresh, i.e. he cannot hear or speak. It says that his actions are valid through gestures; he is no worse than one who became mute, whom we test to see if he can authorize a Get. However, the Gemara says that one who can speak but not hear or vice-versa is like a Pike'ach in every way. It is not clear why the Rambam distinguishes them.

Answer (Kesef Mishneh 2): Surely, one who can speak but not hear or vice-versa is exactly like a Pike'ach. We need not test a Pike'ach! Rather, he is called a Pike'ach as opposed to a total Cheresh. Alternatively, 'one who can speak but not hear is like a Pike'ach in every way' refers to one who can hear a little, but not like Pikchim. This is why the Rambam (Halachah 2) considers one who can speak but cannot hear at all like a total Cheresh.

Poskim

Shulchan Aruch (CM 235:17): If a Cheresh cannot hear or speak, or if he can speak but cannot hear at all, he can buy and sell Metaltelim through gestures, but not land.

Shulchan Aruch (19): If one is hard of hearing but he hears when people talk loudly to him, he is not a Cheresh. He is like a mute.

Shulchan Aruch (121:6): A total Cheresh cannot divorce through gestures if he married when he was Pike'ach, or if a Yevamah fell to him from a Pike'ach brother.

Beis Shmu'el (9): A Cheresh cannot make a Shali'ach. However, we cannot learn from here that Shelichus is not needed to write a Get, since the marriage was not mid'Oraisa.