Shemini (5765)

From the book Mida Tova: Articles on the Hermeneutical Principles by Rabbi Michael Avraham. Translated from Hebrew using gpt-5.4 (reasoning_effort=high, batch API).

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With God’s help — Midah Tovah — Sabbath eve of Parashat Shemini, 5766

Questions

1. Can dayyo refute a kal va-homer (an a fortiori inference), or does it only qualify it?
2. A location-based kal va-homer and a subject-based kal va-homer.
3. Row-based and column-based kal va-homer in relation to dayyo.
4. A new interpretation of the tannaitic dispute regarding dayyo in the passage in Mishnah and Babylonian Talmud, Bava Kamma 24b–25a.
5. Does Rabbi Tarfon really not derive dayyo in a case where it renders the kal va-homer redundant?
6. Why does the Gemara call the redundancy of the kal va-homer its “refutation”?
7. A proposal for a new kind of refuting objection, one that does not undermine the generalization underlying the kal va-homer.

The hermeneutical principles: kal va-homer; dayyo.

“Aaron lifted his hands toward the people and blessed them, and he came down from offering the sin offering, the burnt offering, and the peace offerings. Moses and Aaron came into the Tent of Meeting, and they went out and blessed the people, and the glory of the Lord appeared to all the people.”

— Leviticus 9:22–23

“And Moses and Aaron came into the Tent of Meeting. Why did Moses and Aaron enter together? To teach Aaron about the rite of the incense. Or perhaps he entered only for some other matter. I reason as follows: descent requires a blessing, and entry requires a blessing. Just as descent is part of the service, so too entry is part of the service. From where do we know that entry requires a blessing? It follows logically: if exit, which does not require washing, requires a blessing, then entry, which does require washing, should certainly require a blessing. Or perhaps the reverse: if entry, which does not require a blessing, requires washing, then exit, which does require a blessing, should certainly require washing. No: if you say this of entry, it is because one goes from the ordinary to the holy; will you say it of exit, where one goes from the holy to the ordinary? The reverse inference is therefore cancelled, and we return to the original reasoning: descent requires a blessing, and entry requires a blessing. Just as descent is part of the service, so too entry is part of the service. Why, then, did Moses enter with Aaron? To teach him about the rite of the incense.”

— Sifra, Shemini, parashah 1

A. Summary of Last Year’s Article

In last year’s article we dealt with the hermeneutical principle of kal va-homer through the midrash (rabbinic interpretation) cited above. In the page for Parashat Noah, 5765, we encountered three kinds of kal va-homer: some based on one datum point, whether conceptual or of the type “if two hundred are included, one hundred is included,” and some based on three datum points, that is, quantitative ones. Our midrash is exceptional in that it uses a kal va-homer based on only two datum points.

By presenting the logic of a quantitative kal va-homer as a square table of four laws, we examined the properties of a kal va-homer with two datum points. We saw that in such a case the filled cells, that is, the given data, lie diagonally across from one another, and from them one can construct two kal va-homer arguments, each of which fills a different missing cell in the table. We also pointed out that these two arguments contradict one another.

We noted that when only two datum points exist, there are cases in which it is difficult to define an axis of greater and lesser stringency, because the parameters of one datum point may not be relevant to the other. In other cases, however, the missing data do not result from the absence of such an axis of comparison, but rather from lack of information or uncertainty.

In a situation of missing information, a problem arises: the two possible ways of completing the information contradict one another. We saw that this problem can be solved in two ways:

1. Find a third datum point and complete the table into the form of a regular kal va-homer.
2. Decide between the two conflicting kal va-homer arguments by means of reasoning. This is what the midrash cited above chooses to do.

We further noted that the decision about the relevance of the parameters, that is, about the existence of an axis of comparison, is made by the interpreter even before the kal va-homer itself is carried out. This is the element of reasoning present in every kal va-homer.

Later in the discussion we dealt with a two-datum-point kal va-homer in Babylonian Talmud, Berakhot 21a. Analysis of that derivation shows that an axis of comparison does exist and the parameters are relevant, so the problem there is one of missing information. We described the two ways of completing the table and saw that both appear in the passage.

The conclusion of that discussion was a model of “two transparencies,” according to which the kal va-homer argument is conducted on two planes, one laid over the other. The first is the plane of the biblical text. On this plane we collect the data found in Scripture and determine relations of greater and lesser stringency in light of those data. At this stage, the absence of information about a law is sometimes treated as a positive absence of the law. In the second stage we add data to the table by means of kal va-homer reasoning, and the result may describe different relations of stringency and leniency among the data, since laws that were absent at the biblical stage, and were therefore assumed to be positively absent, are now present. After laying the second transparency over the first, one may arrive at a situation in which the relations of greater and lesser stringency are inverted. But according to our proposal, those relations are determined solely on the basis of the data found on the first transparency, that is, the biblical data. Therefore this is not a refutation of the kal va-homer. In this year’s article we will see another example of this.

We then pointed out that according to Maimonides’ position in the second root, which holds that laws derived by the hermeneutical principles are lighter than ordinary scriptural laws, every kal va-homer of any kind contains an inherent problem: its conclusion refutes its premises. We explained this in light of the two-transparencies model described above.

The conclusion is that kal va-homer is a textual principle, that is, an interpretive tool for Scripture, not for actual Jewish law.¹ The relations of greater and lesser stringency do not reflect real legal relations, but textual ones. These relations are determined solely by the data explicit in Scripture, and not by the actual laws; that is, only with respect to the first “transparency.” For this reason, refutations of those relations of greater and lesser stringency are also determined on the textual plane and not on the plane of actual legal reality.

This also explains the rule that one does not construct a kal va-homer from a law given to Moses at Sinai,² because such a law is not part of the biblical text. We saw that for the same reason one also does not refute a kal va-homer by appeal to a law given to Moses at Sinai.³

B. Applying the “Two Transparencies” Model: Dayyo That Refutes a Kal va-Homer

Introduction

In the article for Parashat Beha’alotekha, 5765, we presented the concept of dayyo le-ba min ha-din lihyot ka-nidon — in brief, dayyo — in light of the passage in Babylonian Talmud, Bava Kamma 25a. At first glance, dayyo is simply a limitation placed on the leverage that one might otherwise have extracted from a kal va-homer. The Bava Kamma passage is highly difficult and complex, but we will nevertheless try to understand some of its basic elements, and through them the mechanism of dayyo as well. We shall see that dayyo, at least in certain situations, constitutes a refutation, on the second transparency, of the hierarchical relations that arise from the first, that is, the textual transparency.

What follows enters the Bava Kamma passage in some detail, and following the argument will require concentration. We believe the effort will be worthwhile, because what is offered here is a proposal that lays out an almost complete picture of the logical aspects of this difficult passage, with which we have been struggling for quite a few years. We suggest that readers, including those already familiar with the passage, follow the discussion with a table such as the one below in front of them throughout the reading. We will speak of row-based and column-based kal va-homer, of subjects and locations, and the reader will be able to see the meaning of the claims by looking at the table.

The Term Dayyo

The Gemara in Babylonian Talmud, Bava Kamma 25a, and in parallel passages, derives the principle of dayyo from the verse concerning the command that Miriam be shut up for seven days, Numbers 12:14:

“The Lord said to Moses: If her father had but spit in her face, would she not bear her shame seven days? Let her be shut up outside the camp seven days, and afterward she may be brought in again.”

If Miriam’s father had spat in her face, she would have had to endure shame for seven days. If so, if the Holy One Himself, blessed be He, spits in her face, she should certainly endure shame for fourteen days, since that is a greater disgrace. But dayyo le-ba min ha-din lihyot ka-nidon: it is sufficient for that which is derived by inference to be like the case from which it is derived. Therefore, even when the Holy One spits in her face, she need be shut up only for seven days.

The assessment of greater and lesser stringency led us to assume that the derived case is more severe than the source case, and therefore that she should endure shame for fourteen days. This is a leveraging of the law of the source case, which speaks of seven days. But dayyo limits that leverage, because we have no proof for imposing an obligation greater than seven days.

Background to the Dispute Between Rabbi Tarfon and the Sages in the Mishnah in Bava Kamma 24b

The scriptural data underlying the discussion are three. In the public domain one is liable for half damages for Horn, and exempt for Tooth and Foot. In the injured party’s property one is liable for full damages for Tooth and Foot. Horn in the injured party’s property does not appear in Scripture.

In last year’s article we presented the data of a kal va-homer and its conclusions in a table. In that format, the scriptural data of the Mishnah in Bava Kamma may be displayed as follows:

Damaging category / domain	Public domain	Injured party’s property
Tooth and Foot	Exempt	Full damages
Horn	Half damages	?

If Horn in the public domain were liable for full damages, this would be an ordinary kal va-homer, and one could infer that Horn in the injured party’s property should also incur full damages. In the article for Parashat Noah we described two arguments that can be used to fill the empty cell:

1. A reasoning based on the rows of the table: from the row of Tooth and Foot we learn that the injured party’s property is more stringent than the public domain, and from there we infer in the second row that if Horn is liable in the public domain, then all the more so it is liable in the injured party’s property. This reasoning rests on the implicit generalization that the injured party’s property is more stringent than the public domain in every respect. The methodologists call this a “location-based kal va-homer,” because it establishes a relation between locations in which legal events occur — the public domain and the injured party’s property — rather than between the legal subjects themselves.
2. A reasoning based on the columns of the table: from the public-domain column we learn that Horn is more stringent than Tooth and Foot. We then move to the second column and infer that if Tooth and Foot are liable in the injured party’s property, Horn is certainly liable there as well. This is called by the methodologists a “subject-based kal va-homer,” because it deals with legal subjects — in this case, categories of damage: Tooth, Foot, and Horn.

So far we have described what the kal va-homer would look like in a symmetrical case. The problem that arises in the Mishnah stems entirely from the fact that this is an asymmetrical kal va-homer, that is, the datum in the lower-right cell is not identical with the datum in the upper-left cell.

Such a situation creates asymmetry with respect to the missing datum as well. If we use the first reasoning, the location-based kal va-homer, the conclusion will be that Horn in the injured party’s property is liable for at least half damages, because this inference is based on the generalization that the injured party’s property is more stringent than the public domain, and Horn in the public domain pays half damages. If, by contrast, we use the second reasoning, the subject-based kal va-homer, the conclusion will be that Horn in the injured party’s property is liable for at least full damages, like Tooth and Foot in that same domain. That is because this inference rests on the generalization that Horn is more stringent than Tooth and Foot, and Tooth and Foot in the injured party’s property pay full damages.

The Dispute in the Mishnah

The Mishnah in Bava Kamma 24b presents a dispute between Rabbi Tarfon and the Sages regarding liability for Horn damage in the injured party’s property:

“Mishnah. How is it with an ox that causes damage in the injured party’s property? If it gores, butts, bites, crouches, or kicks, then in the public domain it pays half damages; in the injured party’s property Rabbi Tarfon says: full damages, and the Sages say: half damages. Rabbi Tarfon said to them: If in a place where the Torah was lenient with Tooth and Foot in the public domain, exempting them, it was stringent with them in the injured party’s property, requiring full damages, then in a place where it was stringent with Horn in the public domain, requiring half damages, is it not logical that we should be stringent with it in the injured party’s property, requiring full damages? They said to him: Dayyo for what is derived by inference to be like the source case: just as in the public domain it pays half damages, so too in the injured party’s property it pays half damages. He said to them: I do not derive Horn from Horn; I derive Horn from Foot. If in a place where the Torah was lenient with Tooth and Foot in the public domain it was stringent with Horn, then in a place where it was stringent with Tooth and Foot in the injured party’s property, is it not logical that we should be stringent with Horn? They said to him: Dayyo for what is derived by inference to be like the source case: just as in the public domain it pays half damages, so too in the injured party’s property it pays half damages.”

Rabbi Tarfon obligates full damages for Horn in the injured party’s property, while the Sages disagree and hold that only half damages are due. At first glance, the simplest explanation of the dispute would be that Rabbi Tarfon and the Sages derive the missing law by means of the two inferences described above, and thus arrive at different conclusions. But the very wording of the Mishnah already makes clear that this is not entirely correct, and in the Gemara this explanation is rejected altogether.

The reason is that both inferences are relevant, and therefore both must be considered. No one may simply decide which of the two to use and which to reject. For example, if Rabbi Tarfon accepts the principle of dayyo only with respect to one kind of kal va-homer, one can always “rotate” the kal va-homer from a row-based argument to a column-based one, and thereby escape dayyo.⁴ Therefore, in practice, according to Rabbi Tarfon the principle of dayyo would not be applicable here at all.

The Course of the Discussion in the Mishnah

At first glance, the sequence of the Mishnah describes Rabbi Tarfon’s attempts to persuade the Sages by means of the two kal va-homer arguments described above. He opens with the first argument, and we have already seen that the conclusion required by it is that Horn in the injured party’s courtyard is liable for at least half damages. Rabbi Tarfon tries to leverage further: the injured party’s property is more stringent than the public domain, and therefore his conclusion is that if Horn in the public domain pays half damages, then in the injured party’s property it should pay full damages. The Sages, as expected, answer with dayyo. Since we have no proof for imposing a liability of more than half damages, we must stop at that amount. Therefore Horn in the injured party’s property also incurs only half damages.

From the wording of the Mishnah it appears that Rabbi Tarfon accepts this response, and therefore tries the second argument. As we saw, the second kal va-homer necessarily leads to the conclusion that liability for Horn in the injured party’s property is at least full damages, and this seems to be a decisive reply to the Sages. Even if one were to apply dayyo here, the law would stop at full damages and not less. Yet, to our astonishment, the Sages reply here too with the principle of dayyo. They argue against Rabbi Tarfon that even the second argument cannot yield more than half damages.

The conclusion that emerges from the Mishnah is that all the tannaim accept the principle of dayyo, but only in relation to the row-based reasoning. The dispute between Rabbi Tarfon and the Sages is only whether the principle of dayyo can also be applied to a reasoning based on the other direction — in our case, the columns.⁵ We should note that Tosafot, s.v. “I too,” and other early commentators following them, such as Nimmukei Yosef and the Rosh, hold that Rabbi Tarfon says these things only according to the Sages’ own position; in his own view, dayyo should not be applied even in the row-based kal va-homer.

Explaining the Second Dayyo

Applying the principle of dayyo to the second argument is very puzzling. The generalization learned from the laws of the public domain is that Horn is more stringent than Tooth and Foot. That is certainly correct, and no objection has yet been raised against it. We now move to the injured party’s property and discover that Tooth and Foot pay full damages there. If so, Horn, which is more stringent, should certainly pay full damages there as well. Why, then, should the fact that Horn in the public domain pays half damages limit the amount of liability in the injured party’s property?

Rashi, in his commentary on the Mishnah, explains this point with a somewhat obscure sentence:

“Dayyo for what is derived by inference to be like the source case — for after all, were it not for Horn in the public domain, the basis for the kal va-homer would not exist.”

The fact that Horn is liable for half damages in the public domain played a role in producing the generalization that Horn is more stringent than Tooth and Foot. Therefore one cannot ignore that fact at the end of the kal va-homer, and one must qualify the leverage and set the liability at half damages. But as pure logic, this explanation is difficult. If the principle of dayyo is based on the claim that we have no proof for leveraging beyond the liability in the source case, then here the final source case is Tooth and Foot in the injured party’s property, where the liability is full damages. Why, then, should one lower the derived case to less than the source case?

Two explanations may be suggested for the application of dayyo to the column-based kal va-homer:

1. The generalization that Horn is more stringent than Tooth and Foot is valid only up to half damages. The first half of the damages is more stringent than the liability of Tooth and Foot, but not the second half. This is a problematic claim, because one could make such a reservation in any kal va-homer that had been challenged, and thereby preserve it.
2. We divide the derivation into two stages:
   a. learning the very fact of liability for Horn in the injured party’s property;
   b. learning the amount of that liability there.

The generalization that Horn is more stringent than Tooth and Foot is valid only with respect to the existence of liability, but not with respect to the amount of liability, because we see that Horn has a unique feature: even when it is liable in the public domain, it pays only half damages. Admittedly, in the public domain the amount of liability for Horn is higher than that of Tooth and Foot. But elsewhere Tooth and Foot pay full damages, and therefore the generalization that Horn is more stringent than Tooth and Foot cannot be applied to the amount of liability. If so, what we learn from the kal va-homer based on that generalization is only the fact of liability, not its amount. The kal va-homer teaches us that Horn in the injured party’s property is liable to pay; the amount of liability is learned from Horn in the public domain, and therefore stands at half damages.

Description of the Dispute in the Gemara

The picture that emerges from the Mishnah is that Rabbi Tarfon and the Sages both agree to the principle of dayyo, but disagree as to whether dayyo applies only to row-based reasoning or also to column-based reasoning.

Yet in the Gemara it appears that the Mishnah was understood as though Rabbi Tarfon rejected the principle of dayyo altogether. Moreover, unlike the picture that emerges from the Mishnah — in which Rabbi Tarfon is actually the more logical position, because he applies dayyo only to the row-based argument, where it is logically built in — the Gemara challenges Rabbi Tarfon specifically: how can he dispute the principle of dayyo, when it is explicit in the Torah?

The Gemara answers that Rabbi Tarfon too accepts the principle of dayyo, but only in a place where some novelty remains after the kal va-homer — that is, “where the kal va-homer is not thereby refuted” — as in the case of Miriam’s seclusion mentioned above. But where, after applying dayyo, the kal va-homer teaches nothing new at all, there dayyo is not said, and the original leverage remains in force.

In our case, after applying dayyo it turns out that the kal va-homer teaches us nothing at all, because we would know even without the kal va-homer that Horn damage in the injured party’s property is paid at the rate of half damages. How would we know that? Rashi explains as follows, and so too do Maimonides in his Commentary on the Mishnah here and others:

“Here half damages is written, and it also implies the injured party’s property; and if by dayyo you set liability in the injured party’s property at half damages, then the kal va-homer teaches us nothing.”

Rashi explains that the verse in the Torah teaching that Horn incurs half damages does not deal specifically with the public domain, but also with the injured party’s courtyard. Therefore half damages would be learned from the verse itself, even without any kal va-homer. Consequently, once we apply the principle of dayyo, the kal va-homer no longer contributes anything new. In such a case, according to Rabbi Tarfon, one does not say dayyo, and the Sages disagree with him on this point. The conclusion is that, contrary to the explanation we proposed on the basis of the Mishnah, the dispute between Rabbi Tarfon and the Sages is not about applying dayyo to column-based reasoning, but about whether one says dayyo when its application renders the kal va-homer redundant — and this in both of the kal va-homer arguments — or not.

Two Ways to Understand the Column-Based Kal va-Homer After Dayyo

We have seen that in our case, after applying dayyo, the kal va-homer becomes superfluous. Liability of half damages can be learned from the verse as well. If so, the situation is as follows: the verse teaches that liability for Horn is half damages in every type of domain. Rabbi Tarfon tries, by force of the kal va-homer, to show that the verse deals only with the public domain, while in the injured party’s property one must require full damages. In other words, the kal va-homer forces us to depart from the plain meaning of the verse and apply what it says only to the public domain.

After applying dayyo, we show that even the kal va-homer does not force us to restrict the scope of the verse only to the public domain, for in any event it turns out that in the injured party’s property too one pays half damages. Two possibilities therefore arise for understanding the final situation:

1. Horn in the injured party’s courtyard pays half damages because of the kal va-homer together with dayyo.
2. Horn in the injured party’s courtyard pays half damages because of the verse, which remains in its plain sense. The kal va-homer is not needed here at all, and teaches nothing in this regard.

A Conclusion from the Gemara

We noted that Tosafot, s.v. “I too,” explain that Rabbi Tarfon, in his own view, does not accept dayyo even with respect to the row-based argument. The reason is that, according to the Gemara, Rabbi Tarfon does not accept the idea of dayyo in a place where it refutes a kal va-homer, that is, where the application of dayyo makes the kal va-homer unnecessary. This claim is true of the row-based argument as well.

Were it not for the words of Tosafot, one could have said that Rabbi Tarfon accepts the Sages’ rejection of the row-based argument and applies dayyo there because it is logically sound. As we saw, this is what seems to emerge from the plain wording of the Mishnah, and perhaps also from Rabbenu Hananel there. According to this proposal, the tannaitic dispute over whether to apply dayyo where it refutes the kal va-homer concerns only the illogical form of dayyo, namely dayyo with respect to the column-based argument, the second one in the Mishnah. Only there does Rabbi Tarfon say that if the kal va-homer becomes unnecessary, we do not apply dayyo.

Two Difficulties with the Gemara

The principle established by the Gemara is not at all clear. Why, when dayyo renders the kal va-homer redundant, do we not use it? Is there some law given to Moses at Sinai obligating us, specifically here, to employ a kal va-homer? If the kal va-homer is unnecessary, then indeed there is no need for it and we should not use it. Why should such a situation force us to give up dayyo?⁶

The terminology of the Gemara is also strange. Why does the Gemara refer to a situation in which the kal va-homer is superfluous as one in which the kal va-homer is “refuted”? The kal va-homer has not been refuted; it has merely been shown to be without value. Prima facie, the generalizations underlying the two arguments we described — that Horn is more stringent than Tooth and Foot, or that the injured party’s property is more stringent than the public domain — remain valid. But we shall immediately see that this is mistaken.

Two Logical Meanings of Dayyo and a Possible Answer to the Difficulties Above

In the row-based kal va-homer, even after applying dayyo, the generalization remains valid: the injured party’s property is more stringent than the public domain. By contrast, in the column-based kal va-homer, the fundamental generalization is that Horn is more stringent than Tooth and Foot, since this is a subject-based and not a location-based kal va-homer. After applying dayyo, however, it turns out that Horn is not more stringent than Tooth and Foot, because in the injured party’s courtyard Tooth and Foot pay full damages while Horn pays only half damages. If so, the application of dayyo has not only made the kal va-homer unnecessary; it has actually refuted the generalization underlying the kal va-homer. In the row-based kal va-homer, dayyo only limits the leverage and makes the kal va-homer redundant, whereas in the column-based kal va-homer it refutes the very kal va-homer on which it is built.

Let us now recall our earlier proposal, that the dispute between Rabbi Tarfon and the Sages concerns only the column-based kal va-homer, contrary to the view of Tosafot but in better accord with the language of the Mishnah. We can now interpret the Gemara’s phrase, “where the kal va-homer is refuted,” literally: where the kal va-homer is actually refuted, and not merely rendered unnecessary.⁷

The second difficulty now also disappears. We can now say that where the application of dayyo refutes the kal va-homer, then if dayyo is logically compelled we would still apply it. But with respect to the column-based kal va-homer we have seen that dayyo does not follow from inferential logic, and therefore there we do not apply it when it refutes the kal va-homer. That is exactly the situation here, and therefore Rabbi Tarfon argues against the Sages that one should not apply dayyo in such a case.

Supplementing the Approach of Tosafot

Our distinction between two types of dayyo perhaps also opens the way to understanding the view of Tosafot. As noted, according to Tosafot Rabbi Tarfon raises the second kal va-homer, the column-based one, only in order to challenge the Sages on their own terms; in his own view he does not apply dayyo even to the first kal va-homer. If so, the problem according to Rabbi Tarfon cannot be that dayyo refutes the kal va-homer, since dayyo does that only with respect to the second kal va-homer and not the first, where it merely qualifies it. So why, then, does Rabbi Tarfon not apply dayyo in the first kal va-homer? The Gemara itself explains that this is because dayyo refutes the kal va-homer. But according to our understanding of the term “refutes,” as opposed to Rashi and most early commentators, that is not correct. Dayyo does not refute the generalization underlying the row-based kal va-homer, namely that the injured party’s courtyard is more stringent than the public domain.

Rabbi Tarfon explains that one does not say dayyo where it refutes the kal va-homer, and in his view dayyo in the row-based kal va-homer also refutes the kal va-homer. Therefore, according to Tosafot too, Rabbi Tarfon does not say dayyo even in the row-based kal va-homer. The explanation of this rests on a kind of refuting objection to the kal va-homer different from the one usually familiar to us.

In the article for Parashat Noah, 5765, we saw that every refuting objection to a kal va-homer undermines the generalization underlying it. Usually, such an objection presents an example showing that the relation of greater and lesser stringency assumed by the kal va-homer is not always correct. That is not what happens here. The relation of greater and lesser stringency assumed by the row-based kal va-homer, that is, the location-based one, remains intact even after dayyo. The injured party’s courtyard is still at least as stringent as the public domain. But the result of the kal va-homer can be attacked from another direction.

To see this, let us look at the two premises in the first column, that of the public domain. From them there emerges the generalization that Horn is more stringent than Tooth and Foot. Yet the result of the column-based kal va-homer shows us that Horn is less stringent than Tooth and Foot, because in the injured party’s courtyard Horn pays half damages, after dayyo, while Tooth and Foot pay full damages. Thus the result of the kal va-homer stands in opposition to the relation that emerges from the two scriptural laws in the public domain. True, there is no direct contradiction to the generalization of the kal va-homer with which we are presently dealing, namely the location-based one. But the result of that generalization creates a problem with two other scriptural data points. On the one hand, the public-domain data imply that Horn is more stringent than Tooth and Foot. On the other hand, the result of the kal va-homer implies from the laws of the injured party’s courtyard that Horn is less stringent than Tooth and Foot.

We propose that perhaps Rabbi Tarfon argues that in such a situation the first kal va-homer too has been refuted, and therefore one does not say dayyo. It should be noted that if we do not say dayyo and continue to use the kal va-homer, no problem arises at all. In that case Horn in the injured party’s courtyard will pay full damages, and it will be as stringent as Tooth and Foot, in accord with the two public-domain data points. Thus the contradiction between the result of the location-based kal va-homer and the two public-domain data points does not lead us to abandon the kal va-homer, but rather to abandon dayyo. The two public-domain data points are evidence that we may not stop the kal va-homer at half damages. Rather, we must require at least full damages so as not to collide with the public-domain data.

As we explained above, dayyo follows from the fact that the generalization of the kal va-homer directs us to apply to the derived case a rule at least as stringent as that of the source case. How much more stringent? Of that we have no proof, and anything more stringent would still remain consistent with the generalization underlying the kal va-homer. Dayyo says that we must take the minimum possible, because we have no proof for more than that. But now we have seen that in the present case we do have proof that we must apply a rule of greater magnitude than that of the source case.

The View of the Sages

What will the Sages say in response to Rabbi Tarfon’s claim? They can argue against him that we have never found a refuting objection of this type to a kal va-homer. Every refuting objection must directly undermine the generalization underlying the kal va-homer, and this one does not do so. Therefore the apparent contradiction between the result of the kal va-homer together with dayyo and the two public-domain data points admits of a different interpretation: it indeed follows from this that there is no clear hierarchical relation between the categories of damage, that is, between Horn and Tooth and Foot. On the one hand, Horn is more stringent in the public domain; on the other hand, in light of the result of the kal va-homer together with dayyo, it is less stringent in the injured party’s courtyard.

But that is the case with every refuting objection. When one raises such an objection against a kal va-homer, the two data points that served as the basis for the generalization teach us a certain relation of greater and lesser stringency, while the datum supplied by the objection shows us that this relation is not always correct, and therefore it may not be applied sweepingly. Usually a refuting objection leads to giving up the generalization of the kal va-homer, but here the generalization of the kal va-homer has not been refuted. In our case, the subject-based kal va-homer between Horn and Tooth and Foot really is refuted in precisely this way, as we have seen: dayyo directly refutes the subject-based kal va-homer. Thus there really is no hierarchical relation between those two categories of damage. But there still is such a relation between the locations, and therefore one may apply dayyo to the location-based kal va-homer and derive the legal conclusion from it. That relation has not been refuted, and there is no reason not to use it, dayyo included.

This is what the Sages answer Rabbi Tarfon, and therefore he raises the second kal va-homer, the column-based one. He is convinced that there they will have to concede, because there the result of the kal va-homer together with dayyo certainly turns back and refutes the very generalization underlying the kal va-homer itself, namely that Horn is more stringent than Tooth and Foot. In that case, dayyo should be rejected. What do the Sages answer him there?

The View of the Sages: Back to the “Two Transparencies” Model

The solution we proposed above will not help us if we adopt the approach opposed to Tosafot, according to which the dispute concerns only the second kal va-homer, that of the subjects. There Rabbi Tarfon raises the claim that dayyo has refuted the generalization underlying the kal va-homer itself, since after dayyo we see from the laws of the injured party’s property that Horn is not more stringent than Tooth and Foot but less stringent than they are. What do the Sages answer to that?

Moreover, even according to Tosafot themselves the matter is difficult. Rabbi Tarfon raises the second kal va-homer against the Sages on the assumption that there they will not be able to use their previous reply against him. There it is clear that dayyo should be cancelled, since it contradicts the kal va-homer, as we explained above. What do the Sages answer him there?

Thus, whether according to Tosafot or according to those who disagree with them, the second kal va-homer is indeed refuted by the result of the kal va-homer together with dayyo, and it is not clear how the Sages can carry out the kal va-homer and remain with this contradictory result.

There are two ways to explain the view of the Sages:

1. In such a case the kal va-homer is indeed refuted, and apparently we learn the law of half damages in the injured party’s property from the verse and not from the kal va-homer. This does not seem to fit the wording of the passage.
2. The second possibility, which better fits the wording of the passage and is the second possibility suggested above, is that we still learn the law of half damages in the injured party’s property from the kal va-homer and not from the verse, but only with respect to the very existence of liability. The amount of payment is learned from Horn in the public domain, and therefore the rule of dayyo applies to it and limits it to half damages.

According to the second possibility, a difficulty still remains: why does the disruption of the relations of greater and lesser stringency that emerges from the results of the kal va-homer together with dayyo not refute the generalization underlying the kal va-homer? After all, one could object as follows: Horn is different, because in the injured party’s property it is liable only for half damages; can you say the same of Tooth and Foot, which are liable for full damages? It is also difficult to say that the generalization is valid only regarding the fact of liability and not regarding the amount of liability, because then the same move could be made in every refuting objection to every kal va-homer: one could say that the refuted generalization is still true, just not in the domain touched by the objection.

Here, it seems, we are compelled to resort to the “two transparencies” model. According to the Sages, dayyo cannot refute the kal va-homer, because the conclusion produced by dayyo belongs to the second transparency. As we explained, kal va-homer is a textual principle, and therefore the hierarchical relation embedded in it, as well as objections to it, may be drawn only from the biblical data themselves and not from the results of the kal va-homer. Rabbi Tarfon and the Sages disagree over whether the results of a kal va-homer can turn back and refute its premises, that is, over whether kal va-homer is indeed a textual principle.⁸

Footnotes