In the article there (Shemini 5765, no. 26), in order to explain two opposite a fortiori inferences from two data points operating in parallel, you present the “two transparencies” model, according to which the a fortiori inference is judged only by the data explicitly stated in the Torah, without the laws derived from interpretive exposition. Therefore, at the stage of the a fortiori inference, the missing law is used as though it were not there, even though in practice there is another derivation, the second a fortiori inference, that teaches that missing law.
If I understood correctly, that is a very strong and very general claim. And if it is true, it is a bit hard to assume the medieval authorities would not have noticed it. What about the explicit Talmudic discussion regarding something learned through a verbal analogy that then goes on to teach through an a fortiori inference? And one can also find examples of refutations based on something not explicit in the Torah, such as at the beginning of Yevamot: “What about a Nazirite, for he can be released by petition,” even though the release of vows is “flying in the air.”
Why not formulate the rule more narrowly—that two derivations that come together begin together from the data points and teach their conclusions in parallel, but in general one does derive and refute also from products of interpretive exposition?
I seem to recall that some wrote that such a deficient a fortiori inference from two data points is really an inductive paradigm.

I assume you meant that something learned through an a fortiori inference should itself teach through an a fortiori inference.
I no longer remember exactly what I wrote, but I do not think that what is learned from interpretive exposition does not play on the field at all. It can be said that only where the two slots can be filled in contradictory ways do we leave them empty, as though there were a negative law there. Maybe that is what you meant here.
It could be an inductive paradigm, but that is not the formulation in the Talmud. It learns here by a fortiori inference. Otherwise this two-sided a fortiori inference really would always remain only by force of an inductive paradigm.
Now I am thinking that perhaps one can also say that when the law does not appear explicitly in the Torah, it is not as though it does not exist, but rather that it is a weaker law, since what is explicit in the Torah is stronger than what is not explicit. Then one can build an a fortiori inference from a weak law to a strong law, and not from absence to a positive law: if in a place where the Torah did not write that one is obligated in a blessing beforehand, namely food, one is obligated afterward, then is it not logical that in a place where the Torah does write a blessing beforehand, namely Torah study, one should be obligated afterward? Maybe I raised that possibility there; I no longer remember.

You raised that possibility there and rejected it, around note 7: if what is explicit is stronger than what is not explicit, then that gap remains even after the a fortiori inference and can serve as a refutation. For example, in the a fortiori inference you wrote, that Torah study is more stringent than food and therefore we will learn that it requires a blessing afterward, one could refute it as follows: what about food, where a blessing afterward is explicitly stated; that is why it has a blessing afterward. Will you say the same about Torah, where at any rate it is not explicit, so perhaps for that very reason it has no blessing at all? [Though maybe one could say to this that we try to minimize the gaps, and since we found Torah study more stringent than food regarding the blessing beforehand, we will try to minimize the gap also regarding the blessing afterward, so that Torah study will at least be obligated even if it is not explicit. But in any case, if one can make a refutation from the gap between explicit and non-explicit, then presumably one can find many more refutations for derivations found in the Talmud, and that requires serious examination.]

If what is learned from interpretive exposition also plays on the field, then a fortiori as a hermeneutic rule becomes again a logical rule, meaning one that deals with laws as they actually stand, rather than a quasi-textual one that ignores laws if they are not learned directly from the text. I just noticed that in the last note, note 11, you wrote: “It is possible that the question of deriving something already derived is also connected to this topic (see on this the sheet for the portion of Pekudei), but this is not the place.” That is what I asked about, the issue of something learned and then going on to teach, though I did not understand why “it is possible” and not “it is certainly connected” to it. I’ll look this evening at the sheet for Pekudei.

By the way, as usual, every time I happen to deal with interpretive exposition I remember with sadness the initiative you wrote about in one of the issues of “A Good Measure,” to build a segmented and orderly database of all the derashot in rabbinic literature.
Especially if it would include all the topics with all their stringencies and leniencies, and with one query one could find all the refutations, categorized for example by a factual refutation like “for he derives benefit,” or by reasoning like “for it concerns eternal life,” or by an explicit law, or by an interpretive derivation, or by a hermeneutic rule.
In addition, it would be possible to generate all the possible a fortiori inferences that can be made on the basis of the data appearing in the Talmud but that are not actually stated there—for example the a fortiori inference in the above-mentioned issue 26, that a doorpost should be obligated in ritual fringes by a fortiori reasoning from four-cornered garments, which are exempt from mezuzah and obligated in ritual fringes. The answer “that is not relevant” is not so clear to me. There are discussions whether one uses “an irrelevant refutation,” but “an irrelevant a fortiori inference” is something that requires more orderly analysis. And maybe it would also be possible to generate a “common denominator” from a large number of source cases.
The problem is that this is hard, long, blacker-than-black work, and presumably whoever would be occupied with it itself would not have enough tools also to analyze and eat the fruits of the carob orchard he is planting. May I ask, with your permission, whether from your position and your university connections you have tried to interest groups in launching such a project? If I had a jar with nine extra lives, I’d invest two of them in building such a database.

Here you returned to a different problem: the contradiction between the two directions of the a fortiori inference. I am discussing the derivation of each one on its own.
Since I have no connections at the university, I do not know how to get such a thing off the ground.

The possibility that a law that is not explicit is lighter than an explicit law, and therefore makes it possible to derive an a fortiori inference, you rejected, as mentioned, because that would be a refutation of every a fortiori inference in the world: “Such a problem arises in every a fortiori inference… if so, the conclusion of the a fortiori inference contradicts its premises [7].” In another formulation: “The result of the a fortiori inference constitutes a refutation of the inference itself: what about the learned case, where the force of the law in it is weaker, since it comes from interpretation, than in the source case, where it is written explicitly in Scripture.”
Is it itself perhaps the case that from the very fact that this is a refutation of every a fortiori inference, it is proven that this is not a good refutation? That is, this is included in the novelty of the hermeneutic rule of a fortiori inference—that this is not a valid refutation. But it may still be that one can make an a fortiori inference from it. In other words, you are assuming here that what counts as a leniency or stringency for purposes of refutation also always counts as a leniency or stringency for purposes of grounding the axis of hierarchy. Maybe not? I have no reasoning to offer on the matter, but one could say that since, as a refutation, it would refute every a fortiori inference, we learn from the rule itself that this is not a good refutation—but we have only the scope of that novelty, and it still could serve for deriving an a fortiori inference.

Now I understand what you are saying, that is, what I was saying. It is possible that you are right. This is a difficult topic for me.