I'm going to try to do this in two parts this week because the topic is pretty long.
The issur of bal tosif appears both in our parsha and earlier in parshas va'eschanan. The GR"A (quoted by Torah Temima in va'eschanan) suggests that we have two pesukim because there are two different types of bal tosif: 1) creating/adding a new mitzvah which does not appear in the Torah; 2) changing the parameters of a mitzvah that does appear in the Torah and adding to it, e.g. adding an extra parsha in tefillin.
Rashi does not seem to subscribe to this distinction. In fact, he seems to not distinguish between the two pesukim at all. Rashi comments in our parsha (13:1) לא תסף עליו – חמש טוטפות, חמשה מינין בלולב, ארבע ברכות לברכת כהנים and in va'eschanan (4:2) he comments לא תוסיפו – כגון: חמש פרשיות בתפילין, חמשת מינין בלולב, חמש ציציות. The two comments are *almost* word for word the same. I highlight *almost* because every stroke of ink in Rashi makes a difference, and here the slight difference in Rashi's words may in fact point to the difference between the two commands. See Gur Aryeh; the L Rebbe also has a brilliant sicha on this Rashi.
Something like 20 years ago I did a post on a machlokes Tos and Rashba with regards to bal tosif. Tos asks why blowing tekiyos d'meyushav on R"H is not a violation of bal tosif? Aren't we adding to what the Torah requires? Tos is mechadesh that that doing a mitzvah repeatedly is not bal tosif. The Rashba gives what seems a far simpler answer: you can't incur bal tosif when you are fulfilling a takana derabbanan. The Torah gives license to Chazal (lo tasur -- next week's parsha) to make takanos and gezeiros. If you are following what Chazal mandate, there cannot be bal tosif. This is such a fundamental yesod that one has to wonder why Tos did not offer this answer, and for that you can read some of the previous posts.
As to why Chazal are not in violation of bal tosif, the Rambam in Hil Mamrim explains that halacha clearly demarcates between Rabbinic law and Torah law. Since a takana cannot be mistaken to be a Torah law, it is not bal tosif.
This year I want to turn to the conclusion of the Rashba and the issur of bal tigra. Rashba adds that just like a takana to do something cannot be a violation of bal tosif, a takana *not to do something* cannot be bal tigra, e.g. when Chazal mandated that we do not blow shofar on Shabbos, this subtraction from the mitzvah is not a bal tigra problem.
The Turei Even asks two questions on this Rashba:
1) Bal tigra doesn't mean missing doing a mitzvah once; it means erasing a mitzvah permanently from the list of 613 things to do. Were that not the case, then any time someone misses doing a mitzvah they would be in violation of bal tigra. Someone who, for example, neglected to put on tefillin, should get malkos for bal tigra in addition to the bitul aseh. That obviously can't be.
When Chazal said not to blow shofar on Shabbos, they were not erasing the mitzvah of shofar. They were telling you to skip it that one time. Hence, no bal tigra.
2) Turei Even argues that bal tigra doesn't mean failing to do a mitzvah. What it means is failing to do a mitzvah *in a complete way.* The mitzvah is still valid, just incomplete. A classic case (R"H 28b):
דתנן הניתנין במתנה אחת שנתערבו בניתנין מתנה אחת ינתנו מתנה אחת מתן ד' במתן ד' ינתנו במתן ד' מתן ד' במתן אחת ר"א אומר ינתנו במתן ד' רבי יהושע אומר ינתנו במתן אחת אמר לו ר"א הרי הוא עובר על בל תגרע אמר לו ר' יהושע הרי הוא עובר על בל תוסיף א"ל ר"א לא נאמר בל תוסיף אלא כשהוא בעצמו אמר לו ר' יהושע לא נאמר בל תגרע אלא כשהוא בעצמו ועוד אמר רבי יהושע כשלא נתת עברת על בל תגרע ולא עשית מעשה בידך כשנתת עברת על בל תוסיף ועשית מעשה בידך
A korban which requires מתן ד is kosher if you sprinkle only מתן אחת; it just hasn't been performed 100% the way it should have been. In that case, the issue of bal tigra comes up. However, let's say you had a korban and didn't do zerikas ha'dam at all. That's not bal tigra -- that's a pasul korban! That's a bitul aseh of offering korban.
The Turei Even proves his case from the following gemara (Sukkah 31b):
אמר מר לא מצא אתרוג לא יביא לא רמון ולא פריש ולא דבר אחר פשיטא מהו דתימא לייתי כי היכי שלא תשכח תורת אתרוג קמ"ל זימנין דנפיק חורבא מיניה דאתי למסרך
The gemara has a hava amina that if there is no esrog available, take some other fruit so that you at least remember that there is a mitzvah of lulav and will do it properly next year.
Turei Even asks: according to the Rashba, how can you have such a hava amina? Taking only 3 of the 4 minim means failing to fulfill the mitzvah, which according to Rashba bal tigra. You can't violate a lav just so you have a zecher?!
It must be that there is no bal tigra in this case because since you are missing the esrog, you are not fulfilling any mitzvah. It's not just an *incomplete* kiyum; it's a nothing kiyum. When you are missing the mitzvah entirely, that’s' not bal tigra.
This question is not only a problem for the Rashba, but is a problem in shitas Rashi as well. Rashi explains that bal tosif is something like adding a parsha to tefillin; bal tigra is the inverse, i.e. taking a parsha away. That's not an *incomplete* kiyum; it's a nothing, and like the Turei Even proves, there is no bal tigra on doing a nothing.
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