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What to do if one discovered his Tallis is Pasul: 
If one discovered his Tallis in invalid, and he is in an area that there is no embarrassment involved in removing his Tallis, such as he is in a private area, then he must immediately remove his Tallis, whether it is a Tallis Gadol or Tallis Katan, and whether it is Shabbos or a weekday. If however one is in a public area, then there is a difference regarding Shabbos and weekday, public and private area, and during times of prayer in Shul.
A. During the week:
Tallis Katan: If one discovered during the week that his Tallis Katan is invalid, then if he is in a public area, he is not required to remove it. Nevertheless, he must [immediately] hurry and go home [or to somewhere private] and remove it as soon as he arrives. This applies even if he is in the midst of a Mitzvah, such as he is in Shul or in a Beis Midrash. [If one discovered his Tallis Katan in invalid and he is in a private area in which there is no embarrassment involved in removing his Tallis, then he must immediately remove his Tallis.]
Tallis Gadol during Davening: If on a weekday, prior to wearing the Tallis Gadol before Davening, one discovered that his Tallis is invalid, it is forbidden for him to wear it even without a blessing. This applies even if he is in Shul [and is partaking in a Minyan]. [If however one only discovered the invalidation after already wearing the Tallis in Shul for Davening, then if he is embarrassed to remove it in front of those present, he may hurry and leave the Shul, and then remove it. If one is in private, he must always remove the Tallis as soon as the invalidation is discovered. In the event that one’s Tallis is invalid and he is embarrassed to Daven in Shul without a Tallis, he may Daven at home without a Minyan.]
If during the week one discovered his Tallis in invalid and he is in an area that there is no embarrassment involved in removing his Tallis, such as he is in a private area, then he must immediately remove his Tallis, whether it is a Tallis Gadol or Tallis Katan, and whether it is Shabbos or a weekday. If however one is in a public area, then by a Tallis Katan he does not need to remove it right away, and rather is to hurry home immediately [or to somewhere private] and remove it. However by a Tallis Gadol one must to remove it right away even if he is in a public area [unless he is in middle of Davening, in which case if he is embarrassed to remove it in Shul, he may hurry and leave the Shul, and then remove it.]
If the Chazan discovered his Tallis is invalid in middle of Davening what is he to do?
Seemingly, if he is very embarrassed to remove it in middle of Davening, he may continue to wear it until he completes the Davening.
What is the law if in middle of a Drasha one notices that his Tallis is invalid? Must he stop the Drasha to remove it, or may he continue wearing it?
Seemingly he may continue to wear it until he is able to complete his Drasha, if he is very embarrassed to stop in middle of the Drasha.
If there are no Kosher Tzitzis strings available in the city, may one follow the leniencies brought next regarding Shabbos?
Some Poskim rule that if there are no Tzitzis available in the city, then one may follow the Shabbos leniencies explained next. Other Poskim however rule that one may not follow the Shabbos leniencies, and the rules above remain fully intact.
May one give his invalid Tallis as a present to another, or make it Hefker, and then continue to wear it without Kosher Tzitzis?
B. On Shabbos:
While walking in a Reshus Harabim: If one discovered on Shabbos, while within a Reshus Harabaim, that his Tallis Gadol or Tallis Katan contains a Biblical invalidation, then he must remove his Tallis immediately. This applies even if as a result of removing the Tallis the person will be left naked in the public marketplace.
While walking in a Karmalis or area with Eiruv, or public building: If one discovered on Shabbos, while within a Karmalis, that his Tallis Gadol or Tallis Katan contains an invalidation, he does not need to remove it. [Nevertheless, as soon as he arrives home he is to remove it. Certainly this law also applies if one is in a public area that has an Eiruv, or in a Shul or any public setting, that one is not required to remove either his Tallis Gadol or Katan until he arrives home. Furthermore, if one is in the midst of a Mitzvah, he is not even required to rush home to remove it. If however one is in a private area in which there is no embarrassment involved in removing even his Tallis Katan, such as at home, then he must immediately remove the Tallis, whether it is a Tallis Katan or Gadol.]
Tallis Gadol During Davening: If prior to wearing the Tallis Gadol before Davening on Shabbos in Shul, one discovered that his Tallis is invalid, then if he is embarrassed to Daven in Shul without a Tallis, he may wear it without a blessing [and remain wearing it until the conclusion of Davening]. [Certainly if one only discovered the invalidation after already wearing the Tallis in Shul for Davening, he may continue to wear it until the conclusion of Davening.] This however only applies if the invalidation was first discovered on Shabbos; if however one knew of the invalidation before Shabbos, then it is forbidden for him to wear the Tallis on Shabbos. [Likewise, this allowance only applies if he does not have another Kosher Tallis available to wear, and is unable to borrow a Tallis.]
Checking the Tzitzis prior to walking in an area without an Eiruv: One is not required to check his Tzitzis to verify their Kashrus status prior to walking through a Reshus Harabim, so long as the Tzitzis were already checked that day.
If on Shabbos one discovered his Tallis in invalid and he is in an area that there is no embarrassment involved in removing his Tallis, such as he is in a private area, then he must immediately remove his Tallis, whether it is a Tallis Gadol or Tallis Katan. If however one is in a public area that is not Halachicly defined as a Biblical Reshus Harabim, then he does not need to remove it right away, and rather is to hurry home immediately [or to somewhere private] and remove it. This applies even by a Tallis Gadol. If however one is in a Biblical Reshus Harabim, then he must remove his Tallis Gadol/Katan immediately. If prior to Davening one discovered that his Tallis Gadol is invalid, then if he is embarrassed to Daven without a Tallis and he did not know of this invalidation before Shabbos, he may wear it without a blessing.
What is the law if one discovered an invalidation on Yom Tov?
It follows the same as Shabbos, with exception that there is no stringency to require to remove the Tzitzis if one is in a Biblical Reshus Harabim.
Second day in Diaspora: Some Poskim rule one is not to initially rely on the above Shabbos leniencies and is rather to try to give the Tallis as a present to another person, and then re-wear it. If this is not possible then one who chooses to be lenient may do so.
C. Informing another Jew that he is wearing invalid Tzitzis:
In an area that he must remove his Tallis immediately: If one notices that the Tallis [Katan] of another Jew is invalid, then if the Jew is found in an area that if he were to discover the invalidation he would be required to immediately remove it [such as in a Reshus Harabim on Shabbos] then one is not to inform him right away but is rather to tell him to go home right away [and escort him home, and] then tell him when he arrives at home that his Tzitzis are invalid. [Alternatively, one can tell him to go to any private area, such as a public bathroom. The above only applies by a Tallis Katan, however by a Tallis Gadol, since there is no embarrassment involved in removing it, one must always inform his friend right away.]
In an area that he is allowed to walk home and then remove his Tallis: If the Jew is found in an area in which he is not required to immediately remove the Tallis upon discovery of an invalidation, but is required to immediately walk home and remove his Tallis [such as by a Tallis Katan being worn during the week in a public area, or on Shabbos in a Karmalis or other non-Reshus Harabim public area, or a Tallis Gadol while Davening during the week] then the person who discovered the invalidation must inform his friend right away and tell him to quickly go home [or to another private area] and remove it.
 13/4; M”A 13/8
 A disputed invalidation: Some Poskim rule that by a disputed invalidation there is no need to remove the Tzitzis right away or even hurry home, [whether by a Tallis Katan or Tallis Gadol]. This would include cases such as if the tear occurred in the braid, or more than two strings are torn but retain Kdei Aniva. [Kaf Hachaim 13/16] Accordingly, the only cases in which one would be required to remove the Tzitzis right away or return home is if: a) the entire Tzitzis tore off the corner; b) A tear occurred below the braid. C) Two strings tore so close to the first knot near the hole that they do not contain Kdei Aniva even according to Riy. However see Biur Halacha 13/3 “Vedavka” and “Aval”.
 The reason: Although by wearing a Tallis without Kosher Tzitzis he is nullifying a Biblical command, nevertheless, since he is not actively nullifying this Mitzvah with his hands, and rather it is nullified consequently of him not removing the Tallis, therefore he is not required to remove it. The reason for this is because the respect of humanity is so great that we allow it to push off a Biblical prohibition through a passive transgression [Shev Veaal Taaseh]. [ibid] It is implied from Admur both here and in 13/6 that by a passive transgression, this applies even if there is only slight embarrassment [Gnaiy Katan]. However by an active transgression, we never permit to do so due to a slight embarrassment [13/5-6] however perhaps we would permit transgressing actively if there is great embarrassment. [See Tehila Ledavid 13/4]
 The reason: As wearing such a Tallis is a Biblical prohibition. [ibid]
 Admur ibid
 One must remove it immediately even if he does not plan to walk with the Tzitzis and will remain standing. [M”B 12/13]
 The reason: As since we do not wear the Tallis Gadol as a garment but rather for the purpose of a Mitzvah, therefore there is no embarrassment involved in removing it. [ibid]
 13/5; Rama 13/3 [does not mention Davening]
Other opinions: Some Poskim rule that if there will not be a Minyan in Shul one may go to Shul and wear the Tallis without a blessing. [Reb Akiva Eiger on Michaber 13/3; Kaf Hachaim 13/10; See Tehila Ledavid 13/4 that it is implied from Admur that if there were to be a Gnaiy Gadol then one may wear it during Davening] See Piskeiy Teshuvos 13/4 footnote 36-37 for a full discussion on this opinion.
 The reason: Although it is embarrassing to Daven without a Tallis, nevertheless it is forbidden for him to wear it being that it is not such a great embarrassment to sit without a Tallis and we do not allow a small embarrassment to push off a Biblical command of the Torah. [Admur ibid; Taz 13/5; M”A 13/8]
 Derech Hachaim 8/2; Siddur Yaavetz; Ketzos Hashulchan 7/11
Ruling of Admur: Admur ibid implies that during Davening the law takes a greater leniency by a Tallis Gadol, and one is hence not required to remove it right away, until he reaches a private area. This is inferred from a) Here he only mentions refraining from the initial wearing and does not explain what to do if the invalidation was discovered after wearing it. b) Here he writes there is “some embarrassment” associated with not wearing the Tallis during Davening while in 13/4 he states “there is no embarrassment” involved in removing the Tallis Gadol, and hence one must say there is a difference in level of embarrassment between the rest of the day and during a public Davening. [Ketzos Hashulchan ibid footnote 34; Tehila Ledavid 13/3]
Other opinions: Some Poskim rule one is required to remove the Tallis immediately in Shul and may not delay doing so until he exits the Shul. [Tehila Ledavid ibid in implication of M”A 13/8; See Piskeiy Teshuvos 13/4]
 See Reb Akiva Eiger on Michaber 13/3; Kaf Hachaim 13/10
 See Reb Akiva Eiger 13/3 and Kaf Hachaim 13/10 that allows to even wear the Tallis even if he is not the Chazan, if he must Daven in Shul.
 See previous Q&A!
 These leniencies include a) Allowance to even initially wear the Tallis Gadol for Davening in Shul. And b) To not remove the Tallis Gadol when in a public area until one reaches home.
 M”A 13/8; P”M 13 A”A 8; Chayeh Adam 11/35; M”B 13/16; Poskim in Kaf Hachaim 13/12; The Kaf Hachaim ibid extends this so far that he permits one to wear a Tallis Gadol without Tzitzis for Davening in Shul even during the week if no Tzitzis are available at the moment that he needs to Daven, even if they are available in the city.
 The reason: As there is no Biblical prohibition involved in wearing a four cornered garment without Tzitzis in such a case, being that one is Anus, it is similar to Shabbos in which we rule there is no Biblical prohibition against wearing Tallis without Tzitzis. [P”M ibid]
 Taz 13/5; Elya Raba 13/6; Mamar Mordechai 13/5
Ruling of Admur: Admur in chapter 13/6 omits the above ruling of the M”A ibid and writes similar to the wording of the Taz which implies to negate the opinion of the M”A and hold stringently. However in 12/6 Admur rules like the M”A 12/7 that one may be lenient like the Riy if he does not have other Tzitzis available, and the P”M ibid explains the reason behind the M”A ibid ruling is because when one does not have Tzitzis it is only a Rabbinical prohibition, and by a Rabbinical prohibition one may be lenient.
 The reason: As the week is a time that Tzitzis can be tied and hence even if it is not available, one still may not wear it without Tzitzis. [Elya Raba ibid]
 M”A 13/8; Yeshuos Yaakov13/4
 M”B 13/15; Artzos Hachaim 13; based on advice of brother of M”A
 13/4; Implication of Michaber 13/3; Shagaas Aryeh 58; Shaareiy Teshuvah 13/2
 This is defined as all areas that a) do not have an Eiruv and b) Have 600,000 people pass through daily.
 What type of invalidation is included in this: In the event that one only tied Tzitzis to three of the corners, the Tzitzis is Biblically invalid until the fourth Tzitzis is tied. It is Biblically forbidden to wear such Tzitzis in a Reshus Harabim. Likewise, if all four Tzitzis have been tied to the corners and a invalidation occurred in one of the Tzitzis, then even if an invalidation occurred in all four corners of Tzitzis due to tears of strings to a length shorter than Kdei Aniva, nevertheless he may not wear these Tzitzis in a Reshus Harabim. If however all the strings of all the four corners have torn in an invalidating manner and there is not even one string remaining that is of the valid measurement that can be salvaged for use for future Tzitzis, and one must hence remove all four corners of Tzitzis and tie new ones to the corners, then the Tallis with these torn strings is not forbidden to wear in a Reshus Harabim due to the carrying prohibition, as these strings have no importance to him and are hence nullified to the garment. It is however forbidden to wear it due to the prohibition against wearing a four cornered garment without Kosher Tzitzis. If however the Tallis is borrowed, which is not obligated in Tzitzis, then one may do so. [Admur 13/1; M”B 13/3; Chazon Ish 3/29]
 The reason: As it is Biblically forbidden to wear a Tallis that contains invalid strings within a Reshus Harabim, and one who does so is liable for a sin offering. The reason for this is because when the Tzitzis are invalid they do not contain a Mitzvah and are hence considered like a package that one is carrying. Now, although they are attached to the Tallis, nevertheless they are not nullified to the Tallis being that the strings are important in his eyes, as he plans on still using them to make Kosher Tzitzis. In the event that one only tied Tzitzis to three of the corners, these three Tzitzis are considered a “package” being that they are important in his eyes, as he plans to keep them until he ties the fourth Tzitzis. Likewise, even if a tearing invalidation occurred in all four corners of Tzitzis, it is still considered like one is carrying a package, as the remaining un-torn strings are important in his eyes as they are still salvageable for use for other Tzitzis. For this reason, the un-torn strings are not nullified to the Tallis and are considered like a package being that they are now invalid and are not an ornament or accessory for the Tallis, as the Tallis does not need these strings at all. [Admur 13/1]
The law if one plans to remove the strings and tie new ones: Some Poskim rule that if the Tzitzis strings are worn out, and one hence plans on removing them and discarding them, and replacing new strings, then he may continue wearing the Tallis even in a Reshus Harabaim. [Aruch Hashulchan 301/108; SSH”K 18/35
 The reason: As the respect of humanity [Kavod Habriyos] does not push off a negative command of the Torah, as the verse states “There is no wisdom or Tevunah or advice against Hashem.” [ibid]
 13/8; Michaber 13/3
 A Karmalis is a domain in which the carrying prohibition is only of Rabbinical nature. [ibid] This is defined as all areas that a) do not have an Eiruv and b) are public and c) 600,000 people do not pass through it daily.
 Michaber ibid states “Tallis”; Rama ibid adds “even a Tallis Katan”; the M”A 13/7 based on Beis Yosef adds even Tallis Gadol; Admur ibid plainly states Tallis, and then states it applies to every type of Tallis.
 The reason: As carrying in a Karmalis, is only a Rabbinical prohibition and the Sages did not enact their decrees in a situation that will compromise on the respect of humanity. Now, although wearing a Tallis without Kosher Tzitzis is a Biblical prohibition, nevertheless on Shabbos there is no Biblical prohibition involved in doing so and it is rather a mere Rabbinical prohibition, [and hence regarding this Rabbinical prohibition we state that the Sages did not enact their decrees in a situation that will compromise on the respect of humanity]. Now, although a Tallis Gadol is only worn for the Mitzvah, and hence does not contain any embarrassment in its removal, nevertheless since the Sages did not decree for one to remove the Tallis Katan in a Karmalis, they did not desire to differentiate in their ruling between a Tallis Gadol and Katan in a Karmalis, and for this reason a decree was never made to remove garments in a Karmalis. [Admur ibid; M”A 13/7] By a Tallis Katan the actual removal is considered belittling of human dignity and hence even though its removal will not cause one to be left unclothed, it is nevertheless included in the original allowance of Kavod Habiryos. [Rama 13/3; M”A 13/7] Now, although earlier [in 13/4] Admur clearly differentiates between a Tallis Gadol and Katan while in a public area, this is because during the week there is a Biblical prohibition involved in wearing an invalid Tallis, and hence if there isn’t embarrassment involved the Sages must differentiate and cannot permit it. However on Shabbos, since the entire prohibition is only Rabbinical, therefore they can choose to be lenient and rule one never has to remove any invalid Tallis in a public area.
 Michaber ibid
 Obviously, the above law applies even when walking in a public area that has an Eiruv, that one is not required to remove his Tallis, whether Gadol or Katan, being the Sages did not differentiate in their decree of Talleisim as explained above, and the reason the above law was written only regarding a Karmalis is just to teach an added novelty that even by a Karmalis the Sages did not decree to remove even the Tallis Gadol.
 M”B 13/16; Kaf Hachaim 13/15
 Tehila Ledavid 13/4 Upashut!; As the Sages allowance to continue wearing an invalid Tallis on Shabbos is only when the concept of “Kavod Habriyos” is relevant, in which case they did not decree against any Tallis, even if it does not involve Kevod Habriyos. However, when in a private area where the concept of Kavod Habriyos is irrelevant, certainly there is a Rabbinical prohibition involved in continuing to wear the Tallis.
 13/6; Rama 13/3
 The reason: Previously [13/5] it was ruled that if this occurred during the week, one may not wear the Tallis in Shul, as a small embarrassment of not wearing a Tallis does not push away a Biblical prohibition. However on Shabbos this law is more lenient. The reason is as follows: On Shabbos it is Biblically permitted to wear a four cornered garment without Tzitzis. The reason for this is because the positive command to place fringes on the garment only begins after one initially wears the four cornered garment, as the Torah does not state “Don’t wear a garment without Tzitzis”. Now, on Shabbos, since it is forbidden to tie Tzitzis to a garment due to the Shabbos prohibition, therefore one cannot fulfill this command. Thus on Shabbos one is not commanded to tie Tzitzis to the corners, and if he wears a four cornered garment without Tzitzis he does not transgress a Biblical command. However during the weekday, since one is able to tie Tzitzis to the corners, therefore if he wears it without Tzitzis he actively transgresses a Biblical command. Now, although on Shabbos there is no Biblical prohibition involved in wearing a Tallis without Tzitzis, nevertheless it remains Rabbinically forbidden to do so, as by wearing the four cornered garment without Tzitzis one is actively causing himself to nullify the positive command of tying Tzitzis. Nonetheless, the Sages did not enact their decrees in a situation that will compromise on the respect of humanity, even if the matter will only be slightly embarrassing, and hence in the above case one may be lenient. [ibid; M”A 13/8]
 Derech Hachaim 8/2; M”B 13/16 and Kaf Hachaim 13/15 regarding even a Tallis Katan.
 13/7; M”A 13/8
The reason: As he should have fixed the Tzitzis before Shabbos, upon discovering the invalidation. [ibid]
 Pashut; Piskeiy Teshuvos 13/4
 13/3; Rama 13/2
 The reason: As since the Tzitzis were already checked one time that day we establish it in accordance to its Chazaka, as explained in 8/13. [ibid]
 Sheilas Shmuel 1; Kaf Hachaim 13/14
 13/9; M”A 13/8; based on Rama Yoreh Deah 303/1
 However by the Tallis Gadol, since there is no embarrassment involved in its removal, therefore certainly he is to be informed right away. See next footnote
 Tzaruch Iyun as to which case Admur refers to. There are only two cases mentioned in which one is required to remove his Tallis immediately even if he is in public. One case is when wearing a Tallis Gadol in public during the week, and a second case is when wearing a Tallis Gadol or Katan on Shabbos in a Reshus Harabim. Now, if it refers to the former case, Admur 13/4 writes that there is absolutely no embarrassment involved in removing a Tallis Gadol in a public area, and hence why shouldn’t the friend be immediately informed about the invalidation. One must thus say Admur refers to the latter case, and specifically when wearing a Tallis Katan [and there is thus embarrassment to remove it in public] and accordingly it is permitted to not tell the Jew of the invalidation even though he is transgressing every four Amos a Biblical prohibition of carrying on Shabbos.
 The reason: As due to our care for human dignity, it is better not to inform the person than to prevent him from doing a merely accidental prohibition. [Admur ibid; Rama Yoreh Deah 303/1]