In summation, there is a brayta on Pesachim 7a which states:
היה יושב בבית המדרש ונזכר שיש חמץ בתוך ביתו מבטלו בלבו אחד שבת ואחד יום טוב
We analyzed this brayta in terms of two related issues. First, we attempt to discover is the true meaning of the brayta in terms of what is being nullified and when it may be nullifed. Second, we consider whether this brayta contradicts other Tannaitic sources in terms of whether one should leave the study hall in order to physically destroy chametz, and offer two resolutions to the apparent contradiction.
In analyzing the brayta, we declared that there are two laws that this brayta is primarily trying to teach. One is that one is allowed to perform nullification on either Shabbat or Yom Tov even though acquisitions (kinyan) is forbidden on Shabbat and Yom Tov by Rabbinic decree. The other is that if one is sitting in the study hall learning and recalls that he possesses chametz (or something about to become chametz) he should not stop learning but should rather nullify the chametz in his heart.
This brayta is understood by the gemara to refer to nullifying dough which is about to become chametz, rather than actual chametz. Analyzing this brayta from a traditional standpoint, we argued that even so, the brayta either explicitly or implicitly agrees that on Shabbat erev Pesach, one should nullify actual chametz as well, rather than leaving the study hall in order to destroy it physically.
Analyzing this brayta and the surrounding gemara from an academic standpoint, we undermined the setama digmara's basis for reinterpreting the brayta as referring to dough which had not yet become chametz rather than chametz itself. That is, Rava had stated that the reason Rav requires nullification in addition to searching is lest one find a tasty loaf, rather than mere insignificant crumbs. The setama objected that if one finds a tasty loaf, he could nullify it then, and answers that, in fact, nullification is ineffective one the prohibition sets in. We suggest that Rava meant that if one finds the significant tasty loaf on Pesach, he would already have violated the prohibition of owning chametz on Pesach, and so subsequent nullification is irrelevant.
The setama mustered up a statement by Rabbi Eleazar citing Rabbi Yishmael as stating that nullification of chametz after six hours on erev Pesach (the time when the prohibition sets in) is ineffective. It does the same for a statement of Rav Gidel citing Rav that states that betrothal using prohibited chametz from six hours and on on erev Pesachis totally ineffective. We suggested that the true meaning of this statement of Rabbi Eleazar is that even though, because of its extreme status of being prohibited in benefit, it is as if it had no substance and is not truly in his possession (as per Rav Gidel), he still is in violation for owning it on Pesach, for the Torah places it as if it in his possession. Rabbi Eleazar was not explicitly stating that one could not nullify it, but perhaps one could derive from Rabbi Eleazar and Rav Gidel's statements that any transfer of ownership, including hefker, would be invalid. This would effectively declare nullification impossible once the prohibition had set in. However, we pointed out that this would be true only if nullification was equal to declaring something hefker, or ownerless, but whether nullification is the same as hefker is subject to debate by Rishonim. Indeed, that the brayta in question states that one may nullify even on Shabbat and Yom Tov suggests that nullification is not a declaration of hefker but rather works to transform the significant tasty loaf into the equivalent of insignificant crumbs. That the same brayta also talks, at least on the simple level before reinterpretation, of nullifying on Yom Tov (once the prohibition has set in) bolsters the idea that this is what nullification is.
Finally, a statement by Rav Yaakov bar Acha was cited which reinterpreted that brayta. We pointed out the irregularity of an Amora answering to issues that until now were entirely in the (assumed to be later) setama, pointed out the setama's hand in the switch from Hebrew to Aramaic, pointed out how there were extraneous irrelevant details in Rav Yaakov bar Acha's statement, and showed how the diyuq from the words of the brayta to support the idea that was nullified was not yet chametz did not really follow. We suggested that Rav Yaakov bar Acha was answering a different question, and that he was really stating the first part of his statement -- that the case in the brayta was one in which one was specifically sitting before his teacher. We bolstered this with related language in the Tosefta which shows that יושב in the study hall means sitting before one's teacher. This cryptic statement by Rav Yaakov bar Acha is really the main link between the two issues and analyses, for we eventually show how he is attempting to resolve an apparent contradiction between Tannaitic sources.
We then turned to the second issue, of the contradiction in Tannaitic sources. While this brayta in Pesachim 7a stated that one should not leave the study hall to physically destroy chametz, other sources implied otherwise. A Tosefta stated that Rabbi Eleazar bar Tzadok, his father, and presumably Rabban Gamliel, left the study hall in order to physically destroy chametz. The Tosefta derives from this the law, stated in a Mishna (in the third perek of Pesachim) as well, that if one is engaged in an important mitzvah and recalls that he has chametz in his house, he should stop what he is doing and destroy the chametz, assuming he has time, and otherwise should nullify it in his heart. Thus, the Tosefta states, and the Mishna implies, that one should leave the study hall in order to physically destroy chametz. A brayta leading up to that Mishna detailing a similar incident with Rabbi Eleazar bar Tzadok may or may not imply the same.
We then noted that Rav Yaakov bar Acha's statement exactly answers this contradiction. Rav Yaakov bar Acha understands the brayta in Pesachim 7a to refer to someone sitting before his teacher, such that there is an additional obligation and critical aspect. Someone merely learning in study hall (such as Rabbi Eleazar and his father after Rabban Gamliel had left to destroy his own chametz) should stop learning to physically destroy chametz. Someone sitting before his teacher who recalls that he (and not others, who have already destroyed their chametz) still has chametz in his possession should not leave his teacher, but should simply nullify the chametz. We thus have found the question that Rav Yaakov bar Acha was trying to answer.
We then point out that a cryptic and troublesome statement in the Yerushalmi on that same Mishna seems to actually be giving the same answer as Rav Yaakov bar Acha -- that while typical learning has the same law as voluntary action in that one would in all instances stop learning to physically destroy the chametz, the case of one sitting before his teacher or before one greater than him in wisdom is not similarly considered voluntary, and therefore, it follows, one should not leave his teacher but should rather nullify it in his heart.
Finally, we interpret the remainder of the Yerushalmi to provide another resolution to the contradiction in Tannaitic sources. All sources deal with simply learning in the study hall, but the opinion that states that one should leave in order to physically destroy chametz is in accordance with Rabbi Yehuda haNasi who maintains that action in performance of mitzvot has greater priority than learning Torah. Meanwhile, the opinion that states that one should not leave the study hall but should rather rely on nullification is the opinion of the Sages who concluded in a vote in Lud that Torah study takes priority. The clarification by the Sages of Caesaria that this is only when the necessary activity (such as burial of the dead) would be otherwise accomplished is in keeping with staying in the study hall, for removal of the chametz is accomplished by means of the nullification.