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Beit Midrash
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Answer: According to your description, the tzitzit are kosher without reattachment, as we will explain. If the other tzitzit are of kosher length (details are not our focus) and only one string ripped, the tzitzit remain kosher for the following reason. The Shulchan Aruch (Orach Chayim 12:1) cites two opinions on leeway regarding ripped tzitzit strings. The Ri posits that each of the four strings can be "too short" on one of its ends if the other end is the proper length. (Background – each corner of the tzitzit has four strings, which look like eight because both ends of the strings face in one direction. One can know the pairings of the string ends if he, as recommended, was consistent about keeping the same four string ends together on opposite sides of the knots.) Rabbeinu Tam holds that two of the four strings must be complete on both ends. If only one of the eight strings became too short, the tzitzit are kosher according to all.
Bemare Habazak - Rabbis Questions (657)
Rabbi Daniel Mann
571 - Peeling an Egg with Writing on it on Shabbat
572 - Reattaching Ripped Tzitzit
573 - Moral Culpability for Unintentional Actions on Shabbat
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The Taz, though, points out that the connection needs to be made at the right time, because of the concept of ta’aseh v’lo min he’asuy (=tvlmh). This means that regarding mitzvot that require acting to create the object of the mitzva (including sukka or tzitzit), the creating must be significant at the time it was done, as opposed to receiving its significance retroactively. For example, if one tied three sets of tzitzit strings on a three cornered garment (for which there is no mitzva of tzitzit) and later forms a fourth corner, the existing sets of strings are invalid until he redoes them (Shulchan Aruch. OC 10:5). So, says the Taz, if one tied strings together before attaching the tzitzit to the garment, all is fine. However, if one originally attached tzitzit strings that were too short and tied on extensions afterwards, it is pasul because of tvlmh.
What is more complicated is when the tzitzit were fine when they were attached, later strings ripped, and one wants to return them by tying. This seems to depend on a difference of opinion regarding a case of a mitzva that was done properly, lost relevance, and was restored (see Pitchei Teshuva, Yoreh Deah 286:13, regarding mezuzot on a doorway which went in, out, and back into obligation in mezuza). The Ba’er Heitev (OC 12:3) cites our Taz as saying that our case would be fine, but the Sha’arei Teshuva (ad loc.), Eliya Rabba (12:1), and Mishna Berura (12:7) point out that the correct reading of the Taz is that it is pasul.
However, the Mishna Berura (ibid.) points out that if one fixed a string when the tzitzit had enough kosher strings, it is not a problem of tvlmh, and the string apparently counts in your case. The value of fixing can show itself if other strings rip even later. Also, an individual short string is not optimal (see Taz ibid.), and reattaching the string apparently fixes that drawback.
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