Taanit 28B

מאי שנא הלל דדחי דידיה, ומאי שנא מוסף דלא דחי דידיה?

What is different about hallel that it overrides its own watch, i.e., the watch in the morning service, when hallel is recited; and what is different about the additional offering, that it does not override its own watch, of the morning service, but it does override the watch of the afternoon service and the closing prayer?

אמר ליה רב אשי: השתא דלאו דידיה דחי ⁠— דידיה לא כל שכן? אמר ליה, הכי קאמינא לך: לא לידחי אלא דידיה!

Rav Ashi said to him: Now, as the additional offering overrides the watch of the afternoon service, which is not its own, is it not all the more so that it should take precedence over its own watch? Rav Ḥisda said to Rav Ashi: I meant the opposite, as this is what I am saying to you: It should not take precedence over another watch, that of the closing prayer; rather, let it override only its own watch, the one performed in the additional service.

אמר ליה: איכא רבי יוסי דקאי כוותך. דתניא, רבי יוסי אומר: כל יום שיש בו מוסף יש בו מעמד. מעמד דמאי? אילימא מעמד דשחרית, הא תנא קמא נמי הכי קאמר! אלא מעמד דמוסף ⁠— דידיה נמי לא דחי?!

Rav Ashi said to him: There is R' Yosei, who holds in accordance with your opinion, as it is taught in a baraita that R' Yosei says: On any day on which there is an additional offering, there is a non-priestly watch. The Talmud clarifies: To which non-priestly watch is he referring? If we say that R' Yosei means the non-priestly watch of the morning prayer, but the first tanna also said this. Rather, R' Yosei must mean the non-priestly watch of the additional prayer. However, this too is problematic; is it possible that the additional offering does not override even its own watch, during the additional service itself?