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hereThe Gemora (Kiddushin 25) explains that while it is true that Rebbe holds that the water is not required to enter the person’s mouth, it must, however, be a place which is fit for the water to enter (and since a bone was lodged between her teeth, the water could not touch her entire mouth; this invalidated her immersion).
This logic follows the opinion of Rabbi Zeira, who says in regards to a korban minchah: A flour-offering that is fit for mixing (of the flour and the oil of the offering; with one log of oil for sixty esronim of flour, and a maximum of sixty esronim in one pan, perfect mixing is possible), the mixing is not critical to it (and the offering will be valid even without mixing); whereas, a flour-offering that is not fit for mixing (where, the proportions of the mixture were less than a log for sixty esronim or where more than sixty esronim were placed in one pan), the mixing is critical (and the offering will not be valid).
Tosfos asks: If the Torah repeated the halachah of “mixing,” it should be critical to the minchah, and if it did not, why is it necessary for it to be “fit for mixing”?
Tosfos answers: Although it is written many times in the Torah, it is not mandatory for it to be mixed, since it is not written in the language of a commandment. We may only derive that the flour and oil should be fit for mixing.
Tosfos in Niddah writes that none of those verses are extra, for they are all necessary to teach various halachos. If so, they ask: Why is it necessary for it to be “fit for mixing”?
Tosfos answers: Since the Torah was particular that a mixing should be done, it is only logical that it should be fit for mixing, for otherwise, the mitzvah would be negated completely.
The Rishonim similarly ask with regards to immersion: Why is it required that his mouth (or other areas) should be a place where water is fit to enter?
Tosfos answers: It is because it is written: And he shall immerse all his flesh in the water. This would seemingly include even all the hidden areas. However, since we expound the verse “his flesh” to be referring only to the exposed parts of the body, the term “all his flesh” teaches us that all parts must be fit for the water to enter.
Evidently, Tosfos holds that this halachah is a Biblical requirement. Other Rishonim hold that it is only a Rabbinical obligation.
Tosfos in Niddah asks: Why isn’t there a requirement at least l’chatchilah that the water should enter even the hidden areas (the same way there is a halachah that the minchah should l’chatchilah be mixed)?
Tosfos answers: With respect to immersion, there is no logic to mandate that the water should enter even the hidden areas of his body, for the Torah is only interested in the person becoming tahor; since b’dieved he will be tahor anyway (even if the water does not come into contact with these areas), what sense is there to require it in the first place? However, with respect to mixing the minchah, which is a mitzvah, it is understandable that the Torah desires that the minchah should be mixed, even though it will be valid even if it isn’t.
My Rosh Yeshivah, Rabbi Avrohom Chaim Levin Shlit”a explains this Tosfos in the following manner: When the discussion pertains to a chalos (something taking effect), it is either valid, or it is not. It is not logical to state that in order for something to be effective, the Torah wants it done in this specific manner. However, even if that is done, it is effective anyway. [L’chatchilah and b’dieved cannot be said regarding a Torahdike chalos.] However, when we are discussing a mitzvah, it is possible to say that there are different levels with respect to the fulfillment of the mitzvah. One will fulfill the mitzvah regardless, but it is still preferable to do it in a certain specific manner.