Hello Rabbi,
We have now reached in our learning the exemption from eating matzah during the days of Passover after the first holiday, based on the interpretive principle of “something that was included in a general category and then singled out was singled out not to teach about itself but to teach about the entire category.”
I’m trying to understand the logic of this principle. Why not say that when something left the general category, it alone left the general category? Why does it teach about the whole category—so that all six days become exempt?

After all, in the Talmud there are examples in both directions. Sometimes there is a verse from whose very appearance we learn that the usual rule is the opposite. And sometimes there are situations in which we derive from it a general paradigm. I usually explain that it depends on what we would have said based on reasoning: if based on reasoning we would have made a general paradigm, then we do so. But if the reasoning points in the opposite direction from the verse, then we view it as an exception and derive that the general rule is the opposite.
As for “something that was included in a category and singled out,” this is a case where a general rule is written and then a detail is written that departs from it. Here too there are two interpretive possibilities: either it comes to create an exception, or it comes to teach. Here, however, there are two possible explanations for why we choose this option specifically:
1. This is the agreed-upon rule on the basis of which the Torah was written. The Holy One, blessed be He, gave us this interpretive principle as a law given to Moses at Sinai, in order to tell us that we should interpret it specifically this way and not the opposite. Therefore, in general, such a principle is needed, and we do not derive it from logic.
2. Perhaps here too one can say that the matter depends on what the a priori reasoning says (whether interpretive reasoning or reasoning about the substance of the issue), and that is what determines the direction of the exposition, as I described above. One suggestion for such reasoning here—this would of course be interpretive reasoning—is that since there is a verbal analogy of “fifteenth-fifteenth” (here in the Talmud a different derivation appears, and I didn’t check why; surely the commentators discuss it) which says that the obligation applies only on the first night, it follows that on the other days it is optional. Another possibility, also interpretive, is that “solemn assembly” usually continues and concludes what came before it; it does not start something new. Therefore the preferred interpretation is that it comes to teach.
Of course, according to option 2, it follows that we will not always expound it in the sense that what left the category came to teach. It depends on the reasoning. That is not the case according to option 1.
There is also more to discuss here, because here the exception was not originally in the category, since the verse speaks about six days and not seven. Rashbam indeed writes that the general category is not the first part of the verse but another verse: “For seven days you shall eat matzot,” and now this was singled out from that category.
Now I’m thinking that perhaps the “six days” in the verse brought here are the last six and not the first six (at least after the seventh was singled out to teach, that implies that the first part of the verse refers to the last six, not according to its plain sense). Then the seventh was included among them and was singled out to teach about them that everything is optional. That would explain everything, because there are two ways to read the verse: either the six days are the first ones, in which case the last one was singled out as an exception; or the six days are the last ones, and the seventh was singled out to teach. Since there is another verse that speaks about seven days of eating matzah, and there are sources that exclude the first day (whether through the “fifteenth-fifteenth” verbal analogy or the derivations brought here), the way the Sages chose is to read the six days as the later days.
I wrote all this quickly without checking the commentators there. Of course, all this still requires further analysis.
All the best,
Michi

Regarding the difference between a general paradigm and “from the fact that it needed to say,” Maharsha on Kiddushin 41b, on Tosafot beginning with “it is derived,” suggests something else (perhaps even contrary). If the verse does not introduce anything new—for example, if it says that agency is ineffective—then there is no point in building a general paradigm, and so we derive by “from the fact that it needed to say.” But if it does introduce something new, then we proceed in the usual way with a general paradigm. It seems to me that this is a necessary and sufficient explanation.
If the reasoning you are suggesting we examine is reasoning strong enough in itself to teach the law without a verse, then it is hard to see why a general paradigm is needed, and there would also be a dispute between you and Maharsha. If we are talking about intermediate reasoning, not enough to teach on its own but enough to support the derivation—something like Tosafot Shevuot 22b, beginning with “if you wish”—then it seems there is no practical difference between you and Maharsha.

There is a lot to discuss in what you wrote.
1. First, there is a difference between a situation in which a verse introduces nothing new and a situation in which an exposition introduces nothing new (see 2 below). An exposition that introduces nothing new is not difficult, because there is no question as to why the verse was written. And the exposition also need not be made at all—unless it is a verbal analogy from a free term, in which case there is a difficulty regarding the free term.
2. The same applies to the rest of your remarks. Usually there is no difficulty as to why a general paradigm is needed, because the Torah did not write anything specifically for the sake of the general paradigm. There is no free term in a general paradigm; the verse is written for its own direct meaning. At most one can ask why the Talmud needed to rely on a general paradigm rather than being satisfied with reasoning, but that is not such a difficult question. Even if one can rely on reasoning, if there is a general paradigm the Talmud mentions it for reinforcement.
3. When there is a verse or exposition teaching a law that makes sense only after the derivation—something like what you called intermediate reasoning, though not exactly the same thing; it does not strengthen the derivation but explains it after we have learned it—then I would say that we do make a general paradigm. But if the matter appears to be a novelty, then one does not extend it beyond its novelty, and we do what you called “from the fact that it needed to say.”

So if I understand correctly and systematize the matter—
Maharsha’s test (“the necessity test”) is relevant only to verses with a free term (and then, if the verse repeats the law that preceded it, we derive “from the fact that it needed to say,” whether the reasoning supports it or opposes it); and your test (“the reasoning test”) is relevant only to derivations without a free term. In other words, your approaches complement each other. Is that right?

I’ll quote the sources for convenience.
Tosafot in Kiddushin there: “And if you say, why not derive that agency is effective for consecrated offerings from the slaughtering of Aaron’s bull, from the fact that it needed to say that it had to be by the owners, as it is written, ‘And he shall slaughter his sin-offering bull that is his,’ implying that with other consecrated offerings we do not require the owners.”
Maharsha: “[From the slaughtering of Aaron’s bull, from the fact that it needed to say… implying that with other consecrated offerings…] Tosafot wrote thus. For one cannot derive by a general paradigm that with other consecrated offerings the owners are required from Aaron’s bull, because if so the verse would not have been needed at all. For from where would we ever have thought that the owners are not required? How would we know agency? This requires further study.”

Indeed.
However, my remarks apply specifically to things learned from exposition, and in such cases there is always reasoning at the root of it (for otherwise there is no reason to expound the verse דווקא that way). See the example of “The Lord your God you shall fear” in this article:
http://www.mikyab.com/#!%D7%A2%D7%9C-%D7%A1%D7%91%D7%A8%D7%95%D7%AA-%D7%9E%D7%A9%D7%9E%D7%A2%D7%95%D7%AA%D7%9F-%D7%95%D7%9E%D7%A2%D7%9E%D7%93%D7%9F-%D7%94%D7%94%D7%9C%D7%9B%D7%AA%D7%99/cu6k/5769345b0cf2644549bdacab
The example of Tosafot and Maharsha seems like a derivation from the verse itself and not from an exposition, since it is stated explicitly in the verse that agency is ineffective (though one could force the verse into a different reading). Here there is an issue of a free term if we were to derive by a general paradigm.