The Logic of Kal Va-Chomer: B. Exceptions (Column 736)

With God’s help

Disclaimer: This post was translated from Hebrew using AI (ChatGPT 5 Thinking), so there may be inaccuracies or nuances lost. If something seems unclear, please refer to the Hebrew original or contact us for clarification.

The Logic of Kal Va-Chomer: B. Exceptions

In the previous column I examined the assumption of relevance that hides behind kal va-chomer (a fortiori) arguments. In its light, I explained the logical flaw in an a fortiori pilpul that is based on only two data points. In this column I will look at additional examples in which the Sages themselves perform a kal va-chomer based on two data points.

Exit and Entry

In Parashat Shemini, the Torah discusses the entry of Moses and Aaron into the Tent of Meeting and their exit from it (Leviticus 9:22–23):

“And Aaron lifted up his <hand> hands toward the people and blessed them, and he came down from offering the sin-offering and the burnt-offering and the peace-offerings. And Moses and Aaron went into the Tent of Meeting, and they came out and blessed the people; and the glory of the LORD appeared to all the people.”

Upon exiting they blessed the people, while nothing is stated about blessing upon their entering.

In the Sifra (Shemini, Parasha 1) this serves as the basis to learn that they also had to bless the people upon entering:

“‘And Moses and Aaron came into the Tent of Meeting’—why did Moses and Aaron enter together? To teach Aaron the procedure of the incense. Or perhaps they entered for some other matter? I will reason: Descending (y’ridah) requires a blessing, and entering requires a blessing. Just as the exit (apparently it should read: descent) is of a kind with the service, so too the entry is of a kind with the service. From where do we know that entry requires a blessing? It is logical: If exit—which does not require washing—requires a blessing, then entry—which does require washing—certainly requires a blessing. Or the reverse: If entry—which does not require a blessing—requires washing, then exit—which does require a blessing—certainly requires washing. No! Do not say entry, for there one goes from profane to sacred; but say exit, for there one goes from sacred to profane. The reversal is nullified, and we return to the reasoning: Descending requires a blessing, and entering requires a blessing. Just as the descent is of a kind with the service, so too the entry is of a kind with the service. Therefore, why did Moses enter with Aaron? To teach him the procedure of the incense.”

This argument relies on an additional assumption, namely that entry requires washing. We thus have a table with two data points, just as we saw in the previous column:

Naturally, two possibilities arise for filling the lacunae cells: in the first a fortiori we learn that entry requires a blessing. Immediately thereafter the argument is raised in the opposite direction—that exit requires washing.

As we saw in the previous column, at first glance these two a fortiori arguments contradict each other. In a fortiori #1 we assume as a datum that exit does not require washing; yet the conclusion of a fortiori #2 is that exit does require washing. And conversely: in a fortiori #2 we assume that entry does not require a blessing, while the conclusion of a fortiori #1 is that entry does require a blessing. The conclusion of one undercuts the premise of the other, and vice versa. On the other hand, we saw that even if both cells are filled with “1,” there is no real contradiction, for at most we have two independent generalizations (binyanei av), and the resulting table is indeed consistent.

Ultimately, the Sifra does see a contradiction here and decides between the two opposing possibilities on conceptual grounds: going from profane to sacred (entry) more properly requires washing than going from sacred to profane (exit), and therefore it is clear that entry requires a blessing. Consequently, it is more reasonable to return to the first a fortiori and conclude that entry requires a blessing (and to remain with exit not requiring washing).

Uncertainty vs. Irrelevance

Two empty cells do not always express irrelevance. Sometimes the empty cells result merely from uncertainty, not irrelevance. In the argument about the doorpost and the tassel (tzitzit) discussed in the previous column, it seemed that the problem was irrelevance; however, in our case (exit and entry) the parameters are clearly relevant, and the issue is uncertainty alone. We do not know whether the lacuna the Torah left indicates obligation or exemption. Therefore the Midrash ultimately decides in favor of the first direction: washing is not relevant when going from sacred to profane; hence that cell should clearly be filled with “0.” That is, the Midrash concludes that this cell is not a lacuna (in which case “X” would be appropriate) but should be filled with “0.” Once we assume that exit does not require washing, the case reverts to a regular “middot-based” a fortiori with three data points.

In other words, the Midrash itself takes for granted that relevance exists here. In this it differs from the doorpost case; hence here there arises an initial thought (hava amina) to reverse the a fortiori. For that reason they are unwilling to accept a result of two binyanei av that would fill the entire table with “1,” and in the end they choose only one of the two directions.

This is evident from the very wording of the Midrash. It presents the matter as a doubt about which direction to take and presumes there is a contradiction between the directions. Where the problem is irrelevance, there is no room to formulate both directions (the “reversal”) and to deliberate between them. As we saw previously, in a situation of irrelevance we are not unsure which is correct; we know with certainty that neither is valid.

Can There Be Irrelevance with Three Data Points?

We saw that with only two data points, the natural conclusion is irrelevance. Yet it is possible that there are two data points and still relevance (and then we search for a value to fill another cell). Note that when a third datum is actually found—or when we begin with three data points—it becomes clear that relevance exists. When the upper-right cell is filled with “0,” this indicates an exemption: the rule is relevant to that state, but there is an exemption. Thus, for example, washing upon exit from sacred to profane is certainly relevant; washing makes sense there. But reasoning suggests that there should not be an obligation—since we are descending from sacred to profane and washing is triggered only when ascending to the sacred. The conclusion is therefore an exemption (“0”) rather than irrelevance (“X”).

As we saw, finding the additional information returns us to a regular “middot-based” a fortiori grounded in three data points. In such a case, the variables are generally fully relevant, and the a fortiori can be performed. A case of a three-datum, “middot-based” a fortiori that expresses not lack of knowledge but irrelevance (and thus cannot be performed) is very rare—though in principle possible. For example, if exit were entirely irrelevant to washing while entry were relevant to blessing. In that case, even if the third datum (that entry requires a blessing) were known, one could still not perform an a fortiori to learn that exit does not require washing.

How then can we know whether we are dealing with irrelevance or with lack of information? As stated, only by conceptual reasoning (sevara). Note that as long as we have not determined the nature of the table’s absences (irrelevance or lack of information), we cannot perform an a fortiori. The upshot is that every regular, “middot-based” a fortiori rests on the exegete’s a priori reasoning that relevance exists. This is not a purely technical formalism; it is the product of an a priori conceptual analysis of relevance.

Let us now turn to another example of a two-datum a fortiori.

Another Example of a Two-Datum A Fortiori: Berakhot 21a

There are a few examples in the Talmud of a fortiori derivations based on two data points. One well-known and clear example appears in the Babylonian Talmud, Berakhot 21a. The Gemara states:

“R. Yehuda said: From where is the obligation of Grace after Meals (Birkat Ha-Mazon) from the Torah?—as it is said (Deut. 8): ‘And you shall eat and be satisfied and bless.’ From where is the obligation of the blessing over Torah study before it from the Torah?—as it is said (Deut. 32): ‘For the name of the LORD I will call; ascribe greatness to our God.’”

Up to this point we learn two data points from the Torah: one blesses for food after eating, and one blesses for Torah before learning. The data table is as follows:

It is now natural to try to complete the two missing entries via a fortiori. We have seen that relying on two data points is problematic, and even if one proceeds, one should choose a single direction. Yet here the Gemara proposes to learn both missing entries together:

“R. Yohanan said: We learn the blessing after Torah from the blessing after food by a fortiori; and the blessing before food from the blessing before Torah by a fortiori. The blessing after Torah from the blessing after food: if food, which does not require a blessing before, requires a blessing after—then Torah, which requires a blessing before, surely requires a blessing after. And the blessing before food from the blessing before Torah: if Torah, which does not require a blessing after, requires a blessing before—then food, which requires a blessing after, surely requires a blessing before.”

Apparently, the Gemara assumes relevance between the rows and columns despite having only two data points—just like the a fortiori above. Conceptually, there certainly appears to be relevance between blessings before and after, both for Torah and for food.[1] It is sensible to bless both before and after on both Torah and food. This is likely the reason the Gemara attempts a two-datum a fortiori here. However, in this case there is no conceptual reason that would lead us to choose only one direction (unlike the washing/blessing case). Therefore, the Gemara fills both lacunae. We end up with a table full of “1”s:

Seemingly we received a binyan av table, but the Gemara learned both entries via a fortiori. That is very problematic, for if both are a fortiori arguments, they contradict each other: each assumes the third datum is “0” and concludes that the fourth is “1.” Yet the third datum for the first a fortiori is the fourth for the second, and vice versa.

The Refutation (pircha)

In the end, the Gemara rejects both a fortiori arguments by presenting refutations: a stringency of Torah and a stringency of food:

“There is a refutation: as to food—there is enjoyment; as to Torah—it is eternal life! Moreover, we learned (Mishnah): one blesses after food but does not bless before it! Refuted.”

The resulting table differs from the initial one in two respects: two “refutation” columns are added, and the lacunae are filled with “0” (not “1” as above):

Why are two columns needed, when only one of them (enjoyment) is actually a refutation?

If one wishes to construct a data table like the first two columns, where the hierarchies are reversed (the columns are independent), then necessarily there will be one parameter with respect to which food is more stringent (controlling the blessing after), and another parameter with respect to which Torah is more stringent (controlling the blessing before).

Thus one can say that the Gemara is not truly adding “refutation columns,” but rather defining the model’s parameters for this table: α—enjoyment (present in food but not in Torah), and β—eternal life (present in Torah but not in food). In other words, the table is 2×2 cells, not 2×4 as I drew above:

Note that the lacunae here are filled with “0” and not “1.” In place of the columns I deleted, we can now use the two parameters revealed by the Gemara to write the model for the final table:

Rashi here, however, presents these refutations as two column-refutations applied to the two a fortiori arguments above:

“There is a refutation—of both. When you seek to learn Torah from food, there is a refutation: as to food—there is enjoyment. And when you seek to learn food from Torah, there is a refutation: as to Torah—it is eternal life.”

That is, two column-refutations are indeed required, because both a fortiori arguments are column-type: one was an a fortiori regarding the blessing before, learning food from Torah; the other, an a fortiori regarding the blessing after, learning Torah from food. We refute the first with “eternal life” and the second with “enjoyment.” Such a situation can occur only when the table contains just two data points and we wish to fill the other two.

The Difficulty

Still, a difficulty remains: how could R. Yohanan adopt a double a fortiori based on two data points, when the two stand in direct contradiction? As we saw, each direction seemingly refutes the other. Note that had the Gemara not raised its two refutations, R. Yohanan’s two arguments would apparently have stood.

In light of this, perhaps we can read the Gemara’s own refutations as indicators: the Gemara indeed intends to reject each of R. Yohanan’s a fortiori arguments by means of the other. The rejection of the first is by virtue of the second, and vice versa. However, a rejection by virtue of some rule must always point to a stringency grounded in conceptual reasoning. For example, the rejection that there is no explicit verse requiring a blessing after Torah indicates that Torah is, in this respect, lighter than food. But then we may ask: why is it lighter? That is what the Gemara explains by saying that Torah is “eternal life”—though its intent is to refute the a fortiori.

The problem is that, if so, it is unclear why the Gemara assumes that both a fortiori arguments are refuted. Seemingly it would suffice to refute one and leave the other intact. Perhaps in this case the Gemara had no way to decide which to retain (“since they are balanced, let both fall”), and because neither is certain we have no basis to establish either of the two novel rulings.

A Proposed Resolution: Rashi’s Remark

Rashi seems to sense this difficulty, and therefore he interprets R. Yohanan’s words in a way slightly different from the plain implication:

“‘Which does not require [a blessing] before it’—that is, we have no explicit verse requiring a blessing before food. ‘Requires [a blessing] after it’—as it is written: ‘And you shall eat and be satisfied and bless.’ ‘Torah which requires [a blessing] before it’—as stated above.”

That is, Rashi attributes food’s “leniency” relative to Torah not to an actual exemption from a blessing before eating, but to the lack of an explicit verse mandating it. According to Rashi’s proposal, the entry “0” we placed in a cell as a working assumption does not indicate legal exemption but the absence of an explicit verse. The assumption is that even if, in practice, food is obligated in a blessing before it, the very fact that there is no explicit verse is itself a leniency.[2]

Rashi’s suggestion indeed eases the difficulty, but it does not fully resolve it. On this reading we have filled both cells with “interpretive 0” but “halakhic 1.” Even if food requires a blessing before and Torah requires a blessing after, those obligations are not explicitly written in the Torah (unlike food-after and Torah-before). Hence they are lighter rulings. But halakhically, in both cases, the obligations stand by virtue of the a fortiori. This yields a further difficulty: suppose we put “interpretive 0” in the “food-before” cell and, via a fortiori, fill the “Torah-after” cell (which had “interpretive 0”) with “halakhic 1.” What now? Torah-after still remains lighter than food-after, since the obligation is not written explicitly. In other words, the hierarchy that Torah is stricter than food does not truly hold; and similarly in the opposite direction for the a fortiori about food-before.

A Reformulation of the Difficulty—For Any A Fortiori

In fact, the difficulty amounts to this: the table that results once both of R. Yohanan’s a fortiori arguments are applied is inconsistent. His table looks like this:

Here “(0,1)” indicates the rule exists in principle but only by virtue of a derivation; it is not stated explicitly in the Torah. That is, “0” interpretively (not in the text), but “1” halakhically (by a fortiori).

We can now immediately see that the datum for “food-before” does not support the a fortiori from which “Torah-after” is concluded (at least as framed above): the reason is that, when we seek to learn “blessing after Torah” by a fortiori, we rely on the generalization that Torah is stricter than food; but according to the conclusion, Torah is lighter than food—since, in the end, “after” the obligation to bless is lighter for Torah ((0,1)) than for food (1). Even if the obligation exists, it is not written explicitly; whereas for food-after it is.

Upon further reflection, this problem appears in every a fortiori—even one with three data points. Consider a regular table (Table 1 from the previous column) applied here:

Let us assume for discussion that there truly is no Torah-level obligation to bless before eating (as in fact the halakhah says; blessings of enjoyment are rabbinic). We now perform an a fortiori and learn that there is an obligation to bless after Torah. But again, that obligation is learned via a fortiori and not written explicitly; thus the lacuna cell will not contain “1” but “(0,1).” Therefore, the assumption that Torah is stricter than food, inferred from the left column, does not actually hold regarding “after” blessings (the right column). Recall: “1” is stricter than “(0,1).” The result of the a fortiori does not change this; it actually demonstrates it. After performing the a fortiori, Torah is still not stricter than food; and if so, the a fortiori itself collapses (for the generalization that served as its basis collapses).

Another Proposed Resolution: A Fortiori as a Textual Rule

It seems that to address this we must distinguish two distinct stages in the derivation. When we examine the scriptural data alone, the picture is that the learned rule is stricter than the teaching rule. But according to the conclusion, it turns out to be less strict. Therefore, we must understand a fortiori as a textual rule, not (only) a logical one. When we consider the biblical text alone, the learned rule appears stricter. True, that is not their real halakhic relationship—after the a fortiori the reverse relation also comes into being—but that is the picture reflected in Scripture itself. That textual picture instructs us to derive the learned rule from the teaching rule by means of a fortiori. Once the learned rule is obtained, the true halakhic relationship is revealed (beyond the interpretive-textual one): the teaching rule, explicit in the Torah, is stricter than the learned rule, which arises only by derivation. But it is the relation as reflected in Scripture that governs the a fortiori.

There is no escaping the conclusion that the a fortiori of the middot is a textual device rather than a strictly logical one. Put differently: the “leniency” and “stringency” operative in a fortiori are not the actual halakhic leniencies and stringencies, but those that emerge from the biblical text alone. In other words, the “(0,1)” value in a lacuna cell is not a value between 0 and 1 (like 0.5). It is a pairing of 0 with 1 on two different planes: interpretively it is 0; halakhically it is 1. Cells that express rules stated explicitly in the Torah contain only a single number because there the interpretive and halakhic values coincide. Needless to say, “(1,1)” and “(0,0)” are simply 1 and 0 respectively. Likewise, “(1,0)” is impossible (for what is written in the Torah certainly holds halakhically—aside from pathological cases of a derivation that uproots or overrides a verse).

I will sharpen the picture by means of a diagrammatic description. On this proposal, a fortiori proceeds in two layers: Layer A gathers the relevant biblical data and records them in a table on Transparency A. Layer B performs the a fortiori inference based on those scriptural data. The midrashic inference generates new halakhot. These halakhot emerge from the derivation and are thus not present in Scripture itself; therefore we record them on Transparency B, which overlays the first. The full halakhic picture is the combination of what appears on both transparencies: the scriptural-textual and the midrashic-halakhic. However, refutations of a fortiori or data used by it cannot belong to Transparency B; they belong exclusively to Transparency A.

Answering the Difficulties

This picture answers the two questions raised above:

• The data on Transparency B are halakhically valid, but they are not scriptural. Therefore they cannot be used to refute the hierarchical relations in the table. This is why the result of the a fortiori does not refute the a fortiori itself. The conclusion that Torah requires a blessing after is not a “0.5”-level obligation that would undermine the assumption that Torah is stricter than food; it is a composite “(0,1),” which refutes nothing. In sum: a fortiori is also a tool of scriptural interpretation, not merely a logical inference. For this reason one cannot refute an a fortiori by arguing, “As for what you seek to learn—its halakhah is only derivational.” Such a refutation belongs to Transparency B.
• If we adopt this picture, then the problem with R. Yohanan’s double a fortiori disappears as well. There, too, only the true halakhic relationship would be contradictory, while what governs the a fortiori is the scriptural relationship—not the ultimate halakhic one. Scripturally, ‘Torah-after’ and ‘food-before’ are absent.

Thus, Rashi’s suggestion in Berakhot—to treat the absence as “not explicit in Scripture” rather than as a halakhic leniency—is not unique to a two-datum a fortiori (like R. Yohanan’s). On our approach, in every a fortiori the leniency and stringency are determined by textual appearance (the scriptural plane), not by the final halakhic ruling (the halakhic plane). Rashi’s proposal therefore resolves all the difficulties.

Implications and Sources for This Proposal

According to this, a fortiori is not merely logical reasoning but also a textual-interpretive tool. This can explain why the Torah transmits the a fortiori measure within the broader system of exegetical rules (middot). It is important to note that there are other types of a fortiori, those based on a single datum plus reasoning rather than on three data points (all biblical a fortiori are of this kind—for example: “Behold, the children of Israel have not listened to me; how then shall Pharaoh hear me?”; likewise all “within two hundred is a hundred”-type arguments). The logical reasoning of a fortiori exists in all such inferences. The middot-based a fortiori (the one conveyed with the exegetical rules) is specifically the sort based on three halakhic data points—a textual tool. Therefore, it requires the Torah’s authorization and revelation to legitimize its use. In the former types we rely on their inherent reasoning.

On this approach it is also easy to understand the tannaitic view that we do not perform a fortiori from a law given to Moses at Sinai (halakhah le-Moshe mi-Sinai; see Mishnah Nazir 56b, and this is the halakhic ruling).[3] Since a law of this sort is not written in the Torah, R. Eliezer (in that Mishnah) holds that the a fortiori cannot apply to it, for it is not a general logical rule but a textual-interpretive one whose force pertains only to what is in Scripture. It is likewise explained in several places that one does not refute an a fortiori on the basis of a halakhah le-Moshe mi-Sinai.[4] These principles clearly indicate that the Sages viewed a fortiori as a tool addressing biblical textual interpretation, not necessarily a logical device. This also seems to emerge from the sugya of “learning from what is itself learned,” for if the product of an a fortiori is not part of the biblical text, it makes sense that one should not build an additional a fortiori on its basis (see more in the article Middah Tovah, 2005, Parashat Pekudei), though this is not the place to expand.

There is an interesting source implying that a fortiori is a textual measure rather than a logical one. See the Hiddushei ha-Rashba to Bava Kamma 2b s.v. “aval be-mehubber,” and the explanation in R. Baruch Ber Leibowitz’s Birkat Shmuel, Bava Kamma, §2, and this is not the place to expand. Another source suggesting that a fortiori is not a logical rule appears in the Brisker Haggadah (in the commentary to “Who knows thirteen?”). See my critique there in the article Middah Tovah, 2005, Parashat Bereshit, where I explained the logic of that inference.[5]

A Different Two-Datum A Fortiori: Harlot’s Hire and an Object of Worship (Asherah)

About seven years ago I was asked in a responsum about a sugya in Avodah Zarah 46b. In the course of the discussion I explained to the questioner that this is another two-datum a fortiori:

“Rava said: A fortiori! If a harlot’s hire (etnan), which is permitted when detached for ordinary use, is prohibited when attached for sacred use—since it is written, ‘You shall not bring the hire of a harlot or the price of a dog’—there is no difference whether detached or attached; then an object of worship (ne’evad), which is prohibited when detached for ordinary use, surely is prohibited when attached for sacred use. Rav Huna the son of Rav Yehoshua said to Rava: Or the reverse: If an object of worship, which is prohibited when detached for ordinary use, is permitted when attached for sacred use—as it is said, ‘Their gods are upon the mountains’—and not ‘the mountains are their gods’—there is no difference for ordinary use or sacred use; then a harlot’s hire, which is permitted when detached for ordinary use, surely is permitted when attached for sacred use…”

Here three axes are in play: etnan vs. ne’evad; detached vs. attached; ordinary use (hedyot) vs. sacred use (gavoah). But the last two are really one axis: detached-for-ordinary vs. attached-for-sacred. We have the following table (1 = prohibited; 0 = permitted):

From the Gemara’s perspective, the entries in the left column are not decisive (the question is whether, when the Torah speaks generally, we make no distinction between detached and attached, or whether we may still be speaking only of detached; hence the question marks). The sugya then offers two opposing a fortiori arguments, seemingly like Berakhot regarding Torah and food. But note that it is not quite the same: the first a fortiori seeks to fill “1” in the “object of worship—attached for sacred use” cell; the second seeks to fill “0” in the “harlot’s hire—attached for sacred use” cell. That is, the two a fortiori arguments address different cells in the right column: the first aims to be stringent regarding the object of worship; the second, to be lenient regarding harlot’s hire.

It is no surprise that the sugya subsequently states:

“He said to him: I argue toward stringency, and you argue toward leniency. Between stringency and leniency—we prefer the stringency…”

This is a comparison between an a fortiori toward stringency and one toward leniency, and is thus not our topic. The rule there is: when we have two possibilities—learning to stringency and learning to leniency—we choose stringency. By contrast, in the cases we saw earlier, both a fortiori arguments lead to stringency: one in the upper-right cell and the other in the lower-left cell. There, the stringency/leniency preference is inapplicable, and thus there is no rule telling us which inference to make. That is the subject of this entire column.

Summary and Conclusion

As I explained in response to Tirgitz in a comment to the previous column, in that earlier piece I concluded that when a table has only two data points, we infer that there is no relevance between the variables and therefore no a fortiori may be made. Accordingly, I rejected the a fortiori obligating a doorframe in tzitzit and a garment in mezuzah. In this column we saw that sometimes relevance does exist even with two data points (as with food and Torah, or blessing and washing).

Therefore the overall conclusion is slightly different: when the table has only two data points, the presumption is non-relevance and an a fortiori may not be made—unless one provides conceptual reasoning (sevara) that relevance exists (as with Torah and food). In such a case, the burden of proof lies on the one who claims relevance. By contrast, in a regular a fortiori based on three data points, the starting point is that there is relevance (since the Torah itself assigns a rule regarding both rows or both columns); there the burden of proof lies on the one who claims non-relevance. See more in my discussion with Tirgitz there.

[1] One could discuss whether, from the mere fact that the Sages instituted a blessing before enjoying food, it follows that the obligation is relevant to the state prior to eating. It could be countered that perhaps on the Torah level it is not relevant at all—but that is unlikely. It is reasonable that the difference between rabbinic and biblical obligations is quantitative rather than qualitative; what is wholly irrelevant biblically would not be instituted by the Sages either. Presumably, the Sages merely lowered the threshold of obligation relative to the Torah.

For a discussion of this from a very different angle, see Daniel Weil, “The Logic of the Sages’ Completions and Greek Logic,” Higayon 1. On the contrast between the Sages’ reasoning and Greek logic (in response to that article), see also M. Avraham’s response, “What Is a ‘Chalut’,” Tzohar 2, and this is not the place to expand.

In any case, since here we are dealing with the biblical plane, later rabbinic enactments are not probative. The question is how the Sages themselves reached their conclusions (that relevance exists). This point arises in every rabbinic enactment, for they always legislate where the Torah was silent; therefore they must ask in each case whether relevance exists and the Torah merely set a higher quantitative bar, or whether the Torah indicates irrelevance—in which case there is no room even for a rabbinic ordinance.

[2] See a similar consideration in R. Elchanan Wasserman’s Kuntres Divrei Soferim, §1 n. 20: what the Torah states explicitly is more stringent than what it merely alludes to by exegetical methods. There are several sources for this among the Rishonim as well, and this is not the place to expand.

[3] See numerous sources collected in Encyclopedia Talmudit, s.v. “Halakhah le-Moshe mi-Sinai,” around notes 191–213.

[4] See Encyclopedia Talmudit there (around notes 203–213).

[5] There I also discuss an a fortiori based on a single assumption; in any case that is a logical a fortiori, not a middot-based one. There the relevant plane is assuredly logic.