Certain mitzvos must be done on a specific day, such as the beris milah of a boy who is healthy on the 8th day, or the Qorban Pesach. In these cases, the mitzvah overrides Shabbos. According to Rabbi Eliezer, this includes even preparatory steps before the mitzvah itself, such as carrying the milah knife in a public domain, things that could have been done before Shabbos, but in this instance they were forgotten. The dominant opinion is that only things that must be done on Shabbos itself override Shabbos, The Qorban Pesach is offered the afternoon before Pesach, even on a Shabbos. The question therefore came up toward the end of the Second Beis haMiqdash period:

שאלו להלל הזקן מה לעשות לעם שלא הביאו סכיניהן עמהם. אמר להן הלכה שמעתי ושכחתי אלא הניחו להן לישראל אם אינן נביאים בני נביאים הן. מיד כל מי שהיה פסחו טלה היה תוחבה בגיזתה. גדי היה קושרו בקרניו. ונמצאו פסחיהן מביאין סכיניהן עמהן. כיון שראה מעשה נזכר הלכה. אמר להן כך שמעתי מפי שמעיה ואבטליון. ר’ זעירה בשם ר”א כל תורה שאין לה בית אב אינה תורה.

They asked Hillel the Elder what to do for the masses who did not bring their knives with them [before Shabbos, so that they could have their qorban slaughtered]. He said to them, “I heard the law, but I forgot it. Leave it to Israel — if they are not prophets, they are the children of prophets.” Immediately, someone whose Pesach offering was a lamb buried [the knife] in its hair. A kid — he would tie [the knife] to its horns. And so it turned out that their Pesach offerings brought their knives with them. Once [Hillel] saw the event, he remembered the law. He said to them, “Like this is what I heard from [my mentors] Shemaya and Avtalyon!”

Rav Ze’ieira in the name of Rabbi Eliezer: Any Torah which has no “ancestral home” isn’t Torah.

– Yerushalmi Shabbos 19:1, vilna 86b-87a

This same story is told in a different, and longer, form in the Bavli:

תנו רבנן הלכה זו נתעלמה מבני בתירא פעם אחת חל ארבעה עשר להיות בשבת שכחו ולא ידעו אם פסח דוחה את השבת אם לאו אמרו כלום יש אדם שיודע אם פסח דוחה את השבת אם לאו אמרו להם אדם אחד יש שעלה מבבל והלל הבבלי שמו ששימש שני גדולי הדור שמעיה ואבטליון ויודע אם פסח דוחה את השבת אם לאו שלחו וקראו לו אמרו לו כלום אתה יודע אם הפסח דוחה את השבת אם לאו אמר להם וכי … מיד הושיבוהו בראש ומינוהו נשיא עליהם והיה דורש כל היום כולו בהלכות הפסח התחיל מקנטרן בדברים אמר להן מי גרם לכם שאעלה מבבל ואהיה נשיא עליכם עצלות שהיתה בכם שלא שמשתם שני גדולי הדור שמעיה ואבטליון אמרו לו ר’ שכח ולא הביא סכין מע”ש מהו אמר להן הלכה זו שמעתי ושכחתי אלא הנח להן לישראל אם אין נביאים הן בני נביאים הן למחר מי שפסחו טלה תוחבו בצמרו מי שפסחו גדי תוחבו בין קרניו ראה מעשה ונזכר הלכה ואמר כך מקובלני מפי שמעיה ואבטליון

Our Rabbis repeated: This law [that the Qorban Pesach is to be brought on Shabbos] was hidden from the sons of Beseirah [who headed the Sanhedrin at the time]. One time, the 14th [of Nissan] fell out on Shabbos. They forgot and did not know whether the pesach trumps Shabbos or not.

The said: Is there no person who knows if the pesach pushes aside Shabbos or not?

They said to them: There is one person, who came up from Babylon named Hillel the Babylonian, who served two great men of the generation, Shemaya and Avtalyon, and would know if the pesach offering pushes aside Shabbos, or if not.

They sent and called for him and said to him: Do you not know if the pesach offering pushes aside Shabbos, or if not?

… [Hillel’s proofs that it does override Shabbos based on derashah and logic ellided] …

Immediately they gave him to sit at the head and appointed him their Nasi. He expoinded the laws of Pesach that whole say. He started to accuse them with words. [Hillel] said to them: Who caused to you that I would come up from Babylon and be Nasi over you? The laziness that was within you, that you did not serve these two greats of the generation, Shemaya and Avtalyon.

They said to him: Rebbe, if he forgot and did not bring the knife on Friday, what is [the halakhah]?

He said to them: I heard the law, but I forgot it. Leave it to Israel — if they are not prophets, they are the children of prophets.

The next day, someone whose Pesach offering was a lamb buried [the knife] in its hair. A kid — he would tie [the knife] to its horns. Once [Hillel] saw the event, he remembered the law. He said to them, “Like this is what I heard from [my mentors] Shemaya and Avtalyon!”

– Shabbos 66a-b

Notice that R Zeira’s (as Rav Ze’ira was called in the Bavli) is missing from this version. Instead there is emphasis placed on Hillel’s ability to prove the first law, and so we are left with the contrasting impression: that had Hillel had a proof and no tradition from the past, he would have been equally willing to give an answer on that basis about the knife, just as he did about the slaughtering itself.

(Side-note: The Bavli’s version makes a mussar point. The implication is that Hillel forgot the halakhah as a consequence of his standing in judgment of Benei Beseira and their Sanhedrin, holding himself superior to them for serving his mentors.)

In any case, the two Talmuds are consistent with their respective general approaches. As I wrote on my notes summarizing my first impressions of learning Yerushalmi, “Invoking Tradition“:

I don’t just mean that because there was no Bavli-style shaqla vetarya, the Yerushalmi fell back onto relying more on sources. Rather, in Eretz Yisrael the feeling was that citations and word-of-mouth transmission is more reliable and more meaningful than relying on reasoning. And this mode was more viable for people who were actually living on the same land and using the same institutions as the tannaim. We saw that Rav Yehudah held it was prohibited for his students to move from Pumbedisa to Israel, as the loss of deep reasoned learning was too great. But also, that kind of reasoning was more critical in Bavel, where there was that extra discontinuity.

It is thus unsurprising that the Yerushalmi sees in the story Hillel’s loyalty to the received tradition. In the Bavli’s version, the focus is instead on Hillel’s mastery of the rules of derashah, and his dispute and give-and-take with the Benei Beseirah defending the merits of argument — and only when he cannot convince based on the merits of the argument does Hillel mention having received the ruling from his mentors.

This notion of loyalty to tradition as it was experienced from one’s mentors also appears in the mishnah in relation to Hillel:

הלל אומר, מלא הין מים שאובין, פוסלין את המקוה–שאדם חייב לומר כלשון רבו; שמאי אומר, תשעת קבין.  וחכמים אומרין, לא כדברי זה ולא כדברי זה; עד שבאו שני גרדיים משער האשפות שבירושלים, והעידו משם שמעיה ואבטליון, ששלושת לוגין מים שאובין פוסלין את המקוה, וקיימו את דבריהם.

Hillel says: A full hin of stored water invalidates a migvah — for a person is obligate to speak in his rebbe’s language. Shammai says: 9 qavin. The Sages say: neither like this one’s words nor like that one’s words. Until two weavers entered via the Trash Gate of Jerusalem, and testified in the name of Shemaya and Avtalyon that three lug of stored water invalidate a miqvah. Then the Sages established their words [to the law].

– Edios 1:3

What does the mishnah mean when Hillel adds “for a person is obligate to speak in his rebbe’s language”? Rashi says that Hillel was explaining why he used the measure of a hin rather than one of the older, halachic standard units. Of course, Hillel’s comment only shifts the question back a generation. If we are to expect tannaim to use halachic-standard units, why didn’t Shemaya and Avtalyon do so? The Rambam rejects this expectation, and instead says that Hillel was commenting on pronunciation. Hillel was so concerned with preserving the tradition as experienced, he immitated his mentors’ accent. Shemaya and Avtalyon were converts, and therefore didn’t grow up with Hebrew. Greek has no /h/ sound, so Hillel repeated the words as they said it, “melo ‘in — a full ‘in“. And then had to explain the dropped hei.

While this mishnah is consistent with Rabbi Ze’ira’s comment, “Any Torah which has no ‘ancestral home’ isn’t Torah”, it actually reduces the power of the story as a proof of the point. All agree that Hillel’s fidelity to the words of his rabbeim was unique, that we aren’t expected to imitate the accent in which we heard the words. So, perhaps it is also part of this uniqueness that kept Hillel from trying to deduce a law he wasn’t taught.