Gemara

(R. Zeira): If one attached Tzitzis to a garment that already had Kosher Tzitzis, the latter are valid.

(Rava): Since he transgresses Bal Tosif (the Isur to add to Mitzvos), attaching the second set is not considered an action.

Objection (Rav Papa): If he does not intend to add to the Mitzvah, rather, to use the new Tzitzis and nullify the first ones, he does not transgress Bal Tosif, so attaching the second set is an action!

Sanhedrin 89a: Extra strings disqualify only if the Torah requires tying the strings in a knot.

Rishonim

Rambam (Hilchos Tzitzis 1:15): If one put additional Tzitzis (on the corner, in addition to the initial Tzitzis, if he intended to nullify the first Tzitzis, he unties the first ones or cuts them, and it is Kosher. If he intended to add to the Mitzvah, even if he cut one set, it is Pasul, for when he added he disqualified everything. When he untied or cut one, the rest are Na'aseh (done, i.e. attached) from Asuy (without a Kosher act), for the first Asiyah was Pasul.

Rebuttal (Ra'avad): The Gemara says unlike the Rambam. Rava explained that attaching the latter srings is a Ma'aseh, due to Bal Tosif, so when he cuts the first, it is Kosher. Rav Papa asked that perhaps the person did not intend to add, rather, to nullify. He does not imply that had he intended to add, one of them would be Pasul. In Sanhedrin, we say that if one put four strings on the corner, and added a fifth, if the Torah requires a knot, it was never Kosher. I.e., an addition disqualifies the Mitzvah. There, the extra striing interrupts between the four strings and the knot. According to the opinion that the Torah does not require a knot, it is Kosher! Likewise, if four strings were tied properly, and one added a string and tied it to them, it does not disqualify them, whether he intended to add or nullify.

Kesef Mishneh: The Rambam explains that Rava says that since Bal Tosif applies, putting on the latter strings is called a Ma'aseh, and we do not disqualify due to '"Ta'aseh", not from what is Asuy.' He asked incredulously, since he transgresses Bal Tosif, can you say that attaching the second set is not a Ma'aseh?! Rav Papa said, perhaps the opposite is true! Perhaps when he intends to add, he transgresses Bal Tosif, and attaching the second set is not a Kosher Ma'aseh, and both sets are Pasul! Since the first were disqualified, cutting the latter will not help, for, and this is like Ta'aseh v'Lo Min ha'Asuy. All the more so cutting the first ones does not help, for the latter were initially put on in a Pasul way. We are Machshir putting on extra Tzitzis only when he intended to nullify the first ones. This is a (Kosher) Ma'aseh, for he did not intend to add. The Halachah follows him.

Rosh (Hilchos Tzitzis Siman 6): We expound that one must have two strings of white and two of Techeles. This does not exclude more, like we find that "Arvei Nachal" mandates at least two, but one may increase (Sukah 34a). Rashi (Menachos 41b) says that one may put eight strings on a corner. However, Sanhedrin 88 connotes that adding Tzitzis disqualifies! It seems that from four to eight is permitted, for we find that some Amora'im put four, and some put eight. The Gemara calls these the minimum and maximum (Menachos 42a). Sanhedrin 89a discusses putting more than eight.

Rosh (19): When one put additional Tzitzis on a garment with Kosher Tzitzis, the latter set is Kosher after the first set is cut, even though when they were put on there was no need for them. Alternatively, it is Kosher through the latter strings, even though the first three of them were Pasul when attached. When he attaches the fourth, this nullifies the initial Tzitzis. We do not say that the first three of the latter Tzitziyos are Ta'aseh v'Lo Min ha'Asuy, because when they were attached this was not a Ma'aseh, due to Bal Tosif.

Beis Yosef (OC 10 DH v'Zeh, citing Mahari Avuhav): According to the first version in the Rosh, even though all the latter Tzitzis were not needed, he can nullify the first ones. The second version requires that the fourth was put l'Shem Mitzvah. Even though the first three were not l'Shem Mitzvah, we do not disqualify due to Ta'aseh v'Lo Min ha'Asuy. Since Bal Tosif applies, it is not considered a Ma'aseh. Therefore, if he intended not for a Mitzvah, and afterwards for a Mitzvah, Ta'aseh v'Lo Min ha'Asuy applies. Here, if we will say that the latter Tzitzis are Kosher, it is because they are additions to the Mitzvah, unlike strings attached not l'Shem Mitzvah to a garment without Tzitzis.

Rebuttal (Beis Yosef, ibid.): The Tzitzis need not be put on l'Shem Mitzvah (Rambam Hilchos Tzitzis 1:12). Our Sugya does not discuss intent for the sake of Mitzvas Tzitzis, rather, for a need or not for a need. L'Halachah, the two versions do not argue. They disagree only about the Perush of the Gemara. The first version says that since the four Tzitziyos were added when there were already Tzitzis, they were unneeded. Even so, they are Kosher after he cuts the first. Since Bal Tosif applied, it was not a Ma'aseh. Cutting off the first ones is the Ma'aseh to make the latter ones. The latter version says that even if he attached only three while the first were still there, and the fourth after he cut the first, we would disqualify, if not that Bal Tosif causes that it is not considered a Ma'aseh. Since the versions do not argue l'Halachah, R. Yerucham did not distinguish based on the intent when attaching the fourth. It seems that the Rosh rules like Rava, that since Bal Tosif applies, it is not a Ma'aseh. Rava did not answer Rav Papa's question. He holds that since if he would intend to add it would not be considered a Ma'aseh, cutting the first ones is a Ma'aseh, even if he did not intend to add, rather, to nullify. This is difficult. It is better to say that the Rosh holds that if he intended to add, all agree that it is not a Ma'aseh, and the latter ones are Kosher. If he intended to nullify, all agree that it is not a Ma'aseh, and Ta'aseh v'Lo Min ha'Asuy applies. They argue about Stam. Rava holds that Stam he intends to add, and Rav Papa holds that he intends to nullify. Rava did not answer, for they argue about Sevara (people's intent). The Tur said 'my father explained differently.' The way I explain the Rosh, he should have said 'my father explained oppositely'! Perhaps he is Machshir in every case. If he cut the latter strings, all agree that it is Kosher. Intent to add to or nullify the initial strings does not disqualify them. Perhaps it is Kosher even if he did not cut the latter ones. Rashi says that it is Kosher even if he cut the first strings. This connotes that all the more so it is Kosher before he cut them! However, if he intended to add, it is Pasul until he cuts them. However, it seems that the Rosh, Tur and R. Yerucham disqualify until he cuts them.

Bach (5): Rashi agrees that it is Pasul until he cuts either pair. All agree that Bal Tosif applies when both sets are tied. However, the latter version in the Rosh holds that when he intended for Bitul, it is Kosher only through the latter Tzitziyos.

Kaf ha'Chayim (10:16): The Acharonim rejected the Bach.

Gra (10:12): It seems that it is Kosher no mater which he cuts. The Gemara' primary Chidush is that it is not Ta'aseh v'Lo Min ha'Asuy.

Poskim

Shulchan Aruch (OC 10:6): If one put extra Tzitzis on a garment, if he intended to nullify the first Tzitzis, he cuts the first ones, and it is Kosher. If he intended to add to the Mitzvah, even if he cut one of them, it is Pasul.

Beis Yosef (ibid.): Even though the Rosh is like Rashi, we rule like the Rambam against him, for his reasoning is better, that intent to add disqualifies.

Rema: Some say that it is Kosher in every case. This is primary. Before he cuts the first ones, it is surely Pasul.

Taz (10:9): I say that according to the first version, the latter are Kosher only if he cut the first. The latter version is Machshir even if the first strings are still there, because he nullified them. The Tur, R. Yerucham and Rema rule like the first version. Regardless of the intent, it is Kosher only after he cuts. This is the Ma'aseh. Since putting on the latter strings can cause Bal Tosif until the first ones are cut, and one may not keep it that way, the Ma'aseh is finished until after he cuts the first ones. The Beis Yosef holds that once the latter strings disqualified the first, they do not become Kosher again. The Rambam explains that Rava argues with R. Zeira. It is a Pasul Ma'aseh, due to Bal Tosif. Rav Papa is Machshir, for presumably the person is not a Rasha, and he intended to nullify.

Mishnah Berurah (23): The Rema discusses cutting the first ones, like the Mechaber did, but all the more so the Rosh (whom we rule like) is Machshir if he cut the latter ones.