1) THE NUMBER OF STRINGS OF "TECHELES" AND "LAVAN"
OPINIONS: The Mishnah states that the lack of Techeles on the strings of the Tzitzis does not prevent the Mitzvah from being fulfilled with the white strings, and the lack of white strings does not prevent the Mitzvah from being fulfilled with Techeles.
How, in practice, are the Mitzvos of Techeles and Lavan performed? How many strings of Techeles and how many strings of Lavan are necessary?
(a) RASHI (DH ha'Techeles) explains that the Mitzvah is to have two strings of Techeles and two strings of Lavan. This is also the opinion of TOSFOS (DH ha'Techeles), the ROSH (Hilchos Tzitzis #6), and the MORDECHAI (#939). Their source for this explanation is the Gemara later (42a) that discusses the verse "Gedilim Ta'aseh Lach" -- "make for yourself fringes" (Devarim 22:12). The word "Gedil" in the singular form refers to a combination of two strings. Accordingly, "Gedilim" in the plural form implies a minimum of four strings (two "Gedil"). These are the four strings that are used as Tzitzis for one corner of the garment (when tied and folded over, they appear as eight strings).
However, this shows only that four strings are needed. What is the source that the four strings must be comprised of two strings of Techeles and two strings of Lavan (and not three of one type and one of the other)? Tosfos answers that since the Torah requires both types of strings, it is logical that there should be an equal number of each type unless specified otherwise.
(b) The RAMBAM (Hilchos Tzitzis 1:6) maintains that the Mitzvah of Tzitzis is comprised of four strings of Lavan, with one of the resulting eight strings dyed with Techeles (that is, half of one full string is dyed with Techeles).
The KESEF MISHNEH explains that the Rambam understands that the word "Pesil Techeles" (Bamidbar 15:38) implies that there should be only one string of Techeles (of the eight strings). Indeed, the Rambam himself writes this in a responsum to the CHOCHMEI LUNIL (quoted by the Kesef Mishneh, Hilchos Tzitzis 1:7), in which he details precisely how Tzitzis are to be made. In his letter, the Rambam also notes that REBBI SHMUEL BEN CHAFNI agrees with his view of the number of strings required of each type. (Tosfos cites a Sifri which may be interpreted as support for the view of the Rambam, but Tosfos concludes that the Sifri itself argues with the Gemara here.)
(c) The RA'AVAD (Hilchos Tzitzis 1:6) says that the Rambam is mistaken, and he presents a third explanation. The Ra'avad maintains that out of the resulting eight strings, two are Techeles and six are Lavan (that is, one full string of the original four is dyed with Techeles). This opinion is also cited by the ARUCH (Erech Techeles) in the name of RABEINU DANIEL.
The Kesef Mishneh explains that Tosfos follows the view of the Ra'avad. This seems problematic, though. As noted above, Tosfos quotes the Sifri that states that only one string of Techeles is required. The Sifri could be interpreted either like the Ra'avad's opinion (one full string of the original four is dyed with Techeles) or like the Rambam's opinion (half of one string of the original four is dyed with Techeles), as the Kesef Mishneh interprets it. However, Tosfos clearly states that the Sifri argues with the Gemara here, and thus he rejects the Sifri in favor of the ruling of the Gemara (that two strings of Techeles are required). Why, then, does the Kesef Mishneh say that Tosfos agrees with the Ra'avad?
(In fact, the words of the Ra'avad as quoted by the BEIS YOSEF (OC 11, in the new printing of the Tur) are "two strings are Techeles and two are Lavan." This is exactly like the opinion of Tosfos mentioned above. However, this text appears neither in our edition of the Ra'avad nor in the older printing of the Beis Yosef. According to the older printing, the Beis Yosef must be referring to the Sifri quoted by Tosfos, and not to the opinion of Tosfos himself.)
(d) The SEMAG (Mitzvas Aseh 26) implies that any number of Techeles strings is acceptable. One may use either two strings of Lavan and two of Techeles, or three of Lavan and one of Techeles. The Beis Yosef explains that the Semag does not mean that either option is a valid Halachic option. Rather, since the issue was not relevant to practical Halachah, the Semag did not trouble himself to decide whom the Halachah follows, and thus he simply recorded both opinions. (Y. MONTROSE)
2) A TOTALLY CUT OFF STRING
QUESTION: Rava states that if the strings of Lavan of one's Talis are cut off and the strings of Techeles remain whole, or visa versa, the Talis is still valid. The Gemara explains that the Talis is valid only if the remnants of the strings that were cut off retain "Kedei Anivah" -- "enough to tie" (the Gemara later discusses the meaning of this term). The Gemara later (39a) says that if a string is cut off entirely, the Talis is Pasul.
The RAMBAM (Hilchos Tzitzis 1:18) rules in accordance with the Gemara here. However, the Rambam earlier (1:4) explains that the Mishnah's case in which "the Lavan does not prevent the Mitzvah of Techeles from being fulfilled" is when all of "the Lavan gets cut off up to the corner of the garment, and only the Techeles remains" (which, according to the Rambam's opinion (see previous Insight), means that only one string out of eight remains), and the Rambam rules in that case that the Talis is still valid! How can the Rambam rule that if one string is entirely cut off, the Talis is Pasul, while if all of the strings of the Lavan are cut off, the Talis is valid? (KESEF MISHNEH)
(a) The KESEF MISHNEH answers that the Rambam maintains that it is not the number of strings that determines whether or not the Talis is valid, but rather the number of sets of strings. When the Rambam rules that the Talis is Pasul if one string tears off, he refers to a case in which an entire string fell off before the strings of the Tzitzis were knotted. An entire string consists of one string of Techeles and one of Lavan (when folded over, since half of the string is dyed with Techeles). When all of the Lavan strings are cut off, the Talis is valid because the Techeles remains intact. When all the strings are cut off, then it is necessary to have "Kedei Anivah" on all of their remnants in order for it to be a valid Talis.
(b) Alternatively, the Kesef Mishneh explains that when the Rambam says that the Lavan strings were cut off "until the edge of the garment," he means that there is still "Kedei Anivah" left on all of the Lavan strings. Accordingly, the Rambam maintains that the Talis is valid only when one set has "Kedei Anivah" and the other is intact. Unlike the first explanation of the Kesef Mishneh, if the Techeles is also only "Kedei Anivah," then the Talis is completely Pasul.
(c) The Kesef Mishneh suggests a third explanation. The Rambam understands that there are three different cases with three different Halachos. When all of the strings are partially cut off, "Kedei Anivah" is required in order for the Talis to be valid. When one string is totally cut off, then the Talis is Pasul. If the Lavan strings are only partially cut off but do not have "Kedei Anivah," the Talis is valid as long as the Techeles is intact.
(d) The KEREN ORAH answers that when the Rambam (1:18) says that when one string is totally cut off it is Pasul, he does not mean that the entire Talis is Pasul. He means that although the person no longer fulfills the Mitzvah of whichever type of string (Techeles or Lavan) that was cut off, he still fulfills the Mitzvah of the other type of strings. This is similar to the Rambam's ruling earlier (1:4), where he states that the Lavan does not prevent the Mitzvah of Techeles from being fulfilled. He means that only the Mitzvah of Lavan remains unfulfilled (unless the Lavan strings have "Kedei Anivah.")
(e) The ARUCH HA'SHULCHAN (OC 12:7) explains the words of the Rambam in a very simple manner. The question on the Rambam's ruling is based on the premise that the words "totally cut off" mean that the strings are totally cut off from the knot. This, however, is not the correct definition of these words. The Rambam explains, like RASHI (39a, DH me'Ikaro), that "totally cut off" means from the place where the Tzitzis are connected to the garment. Accordingly, there is no contradiction in the Rambam's rulings. The Rambam rules that if one string was cut off from the garment, the Talis is Pasul. If the Lavan strings are cut off until the knot, but not from where they meet the garment, the Talis is valid as long as the Techeles strings remain. If all of the strings are cut off until the knot, the Talis is valid if there remains at least "Kedei Anivah." (Y. MONTROSE)