Beshalach (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 — Eve of the holy Sabbath, Parashat Beshalach 5766

Questions

1. Hekkesh (an inference from juxtaposition within a verse) and gezerah shavah (a verbal analogy based on identical expressions) as two building blocks of midrashic analogy.
2. Finding long-range textual correlations as a two-dimensional graph unfolded by hekkesh and gezerah shavah.
3. A brief note on logical sorites.
4. A midrashic sorites, and its relation to the logical sorites.
5. Why does the validity of an exegetical derivation composed of many hermeneutical principles not decline with the length of the chain?
6. How does all this relate to the issue of deriving one learned law from another learned law?

The Principles

• gezerah shavah
• hekkesh
• semukhin (inference from adjacency)
• kal va-homer (a fortiori inference)
• binyan av (derivation from a paradigmatic case)
• kal va-homer

A. Summary of Last Year’s Article

“See that the Lord has given you the Sabbath; therefore He gives you on the sixth day bread for two days. Let every man remain in his place; let no man go out from his place on the seventh day.”

— Exodus 16:29

“Where are these two thousand cubits written? For it was taught: ‘Let every man remain under him’—this teaches four cubits. ‘Let no man go out from his place’—this teaches two thousand cubits. From where do we know this? Rav Hisda said: We learn ‘place’ from ‘place,’ and ‘place’ from ‘flight,’ and ‘flight’ from ‘flight,’ and ‘flight’ from ‘boundary,’ and ‘boundary’ from ‘boundary,’ and ‘boundary’ from ‘outside,’ and ‘outside’ from ‘outside.’ For it is written: ‘You shall measure outside the city, on the eastern side, two thousand cubits,’ and so forth. But let us learn from ‘from the wall of the city and outward, one thousand cubits’! We derive ‘outside’ from ‘outside,’ and we do not derive ‘outside’ from ‘outward.’ What practical difference is there? But the school of Rabbi Ishmael taught: ‘And the priest shall return, and the priest shall come’—this returning is the same as this coming! That applies only where there is nothing else more similar to it; but where there is something more similar, we learn from what is more similar.”

“Two thousand cubits in a circle. But as for Rabbi Hanina ben Antigonus—what is his position? If he accepts gezerah shavah, then ‘corners’ are written; and if he does not accept gezerah shavah, from where does he get two thousand cubits? In truth, he does accept gezerah shavah, but here it is different, because Scripture says: ‘This shall be to them the open land of the cities’—to this you give corners, but you do not give corners to those resting on the Sabbath. And the Rabbis? It was taught: Rav Hanania says, ‘Thus shall all those resting on the Sabbath be.’”

— Babylonian Talmud, Eruvin 51a

“‘Let no man go out from his place on the seventh day’—these are the two thousand cubits that are permitted on the Sabbath for walking in any direction from the place where one spent the Sabbath. How is this implied? Hisda said: We learn ‘place’ from ‘place.’ It is written here, ‘Let no man go out from his place,’ and it is written there, ‘And I will appoint you a place to which he may flee’ (Exodus 21:13). And ‘place’ we learn from ‘flight,’ for in that same verse it says, ‘to which he may flee.’ And this ‘flight’ we learn from ‘flight,’ for it is written there, ‘If the manslayer at all goes out beyond the boundary of his city of refuge, to which he fled’ (Numbers 35:26). And this ‘flight’ we learn from ‘boundary,’ for in that same verse it says, ‘the boundary of his city of refuge.’ And this ‘boundary’ we learn from ‘boundary,’ for it is written there, ‘and the blood-avenger finds him outside the boundary of his city of refuge’ (ibid., 35:27). And this ‘outside’ we learn from ‘outside,’ for it is written there, ‘You shall measure outside the city, on the eastern side, two thousand cubits’ (ibid., 35:5). Thus we have learned that the Sabbath boundary is two thousand cubits. And if you say that these two thousand cubits were stated regarding the cities of refuge—there is also written, ‘From the wall of the city and outward, one thousand cubits round about’ (ibid., 35:4). And if you say that the two thousand cubits were for fields and vineyards, and the one thousand cubits were for open land, even so we learn from here the Sabbath boundary by hekkesh and gezerah shavah. A hekkesh is where the matters are not similar to one another, neither in writing nor in pronunciation, but since both are written in one verse we learn one from the other—for example, ‘place’ from ‘flight,’ and ‘flight’ from ‘boundary,’ and ‘boundary’ from ‘outside.’ A gezerah shavah is two matters that resemble one another, and although they are not in one verse we learn one from the other—for example, ‘flight’ from ‘flight,’ and ‘boundary’ from ‘boundary,’ and ‘outside’ from ‘outside,’ and so throughout the Torah. And the Rabbis said: Kal va-homer is given to every person to expound; gezerah shavah and hekkesh were not given to be expounded, except where the Sages expounded them according to the tradition in their hands.”

— Sechel Tov (Buber edition), Exodus 16, on “let no man go out”

In last year’s article we dealt with the issue of the Sabbath boundary. The Talmudic passage cited above asks from where we derive the rule of two thousand cubits and offers a source in the form of a complex gezerah shavah, built as a seven-link chain alternating between gezerah shavah and hekkesh.

We saw that the gezerah shavah from which the rule of two thousand cubits is learned is disputed: some maintain that it is rabbinic, while others maintain that it is a full Scriptural derivation. Even Rabbi Akiva, who holds that the Sabbath boundary of two thousand cubits is a biblical prohibition, learns the measure of two thousand cubits from a different derivation. It therefore seems that even in his view this passage is only an asmakhta (a scriptural allusion used as support).

We noted that the claim that a given derivation is an asmakhta requires explanation. If it met all the criteria for a valid derivation, why would we not derive law from it? We are forced to assume that there is some flaw in the derivation if it is to be classified as an asmakhta.

What is the structural flaw in this derivation? The commentators tie it to the biblical contexts. The author of Torah Temimah explains that the plain sense of the verses deals with gathering the manna, not with going beyond the Sabbath boundary. That is a difficult claim, for on that basis we ought to derive the law of two thousand cubits with respect to gathering manna. Beyond that, since when does the fact that an exegetical derivation contradicts the plain sense deprive it of halakhic (legal) standing? The author of Torah Shelemah (Beshalach, supplements, sec. 22) explains that this derivation is an asmakhta because the same verse is also used to derive the prohibition of carrying; this may depend on disputes among the medieval authorities. By contrast, we proposed an explanation that arises from the very character of the derivation.

The derivation opens with the question of the source for the measure of two thousand cubits and answers it with a long derivation built from seven successive stages of gezerah shavah and hekkesh in alternation. The midrash (rabbinic exegetical text) Sechel Tov cited above details them: the gezerah shavah “place”-“place”; the hekkesh “flight”-“place”; the gezerah shavah “flight”-“flight”; the hekkesh “boundary”-“flight”; the gezerah shavah “boundary”-“boundary”; the hekkesh “outside”-“boundary”; and the gezerah shavah from Sabbath limits to the Levitical cities.

We pointed out that all the stages in the chain apparently do nothing more than transfer semantic meaning from place to place, in gezerah shavah, or from concept to concept, in hekkesh. No legal parameter passes through the links of hekkesh or gezerah shavah that make up the chain. The legal parameter passes only in the final gezerah shavah: the two thousand cubits in the Levitical cities teaches that in Sabbath limits as well the distance is two thousand cubits. We noted that, in its external form, such a structure resembles the formula known in Greek logic as a sorites.¹

We noted that there are two ways to understand the logic of this midrashic chain:

1. This is a chain in which each link is concerned with transferring semantic meanings. The substantive legal inference exists only between the endpoints: from the first stage, the Levitical cities, to the last, the Sabbath boundary.
2. We are learning a series of gezerah shavah and hekkesh in alternation. Every time we compare a word to an identical word elsewhere, that is a gezerah shavah; in between, we connect two different words that appear in the same context, that is, the same verse, by means of a kind of hekkesh.

We noted that this may be the difference between the sources. Sechel Tov, which details every stage, shows that there is a full derivation at each stage, whereas the Talmud, which describes everything in one sentence, treats the whole structure as one comprehensive derivation.

We then pointed to a reflection of these two approaches among the medieval commentators. Rashi understood the derivation in the first way, whereas the Meiri there and Tosafot on Sotah apparently understood each link in the chain as a complete derivation. The practical difference is clear: according to Rashi, nothing can be inferred about the two thousand cubits from any of the intermediate stages; only the endpoints matter. According to the Meiri and Tosafot, however, one can infer the two thousand cubits even at the intermediate stages.

We observed that according to Rashi’s approach it is difficult to understand what justifies such a semantic transfer if there is no substantive similarity, and if so these are not genuine derivations of gezerah shavah and hekkesh. This may be why some regard the whole structure as an asmakhta. According to the Meiri and Tosafot, it appears plainly to be a full derivation. We also raised the possibility that these are full derivations even according to Rashi; only because we have no application for them until the final stage do we use them merely as semantic intermediaries.

With respect to the laws of Sabbath limits, we pointed out that perhaps Rashi and the Meiri disagreed over whether the foundation of the law is the obligation to remain in the same place, a law about a state of affairs, or whether it is a prohibition on going out beyond one’s place, a law about an action, like the prohibited labors of the Sabbath.

B. A Midrashic Sorites

Parallel discussions

Another midrashic sorites appears in Babylonian Talmud, Pesachim 7b, regarding the source for the obligation to search for leaven by candlelight.² It should be noted that there too the teaching is presented by Rav Hisda, just as in our case. Moreover, in terms of the structure of the derivation and its meaning, it seems there to be a genuine derivation and not an asmakhta. Most of the links in that derivation appear, at least on the face of it, to be full derivations in their own right, and it even seems possible to apply the law learned from the derivation at the intermediate stages; for that reason, it is less plausible there to interpret the structure as mere semantic transfer. On the other hand, the Talmud there on 8a makes clear that it is only a hint, which is unsurprising, since the rule is rabbinic. We could not, within the framework of last week’s article, examine that parallel discussion, and we leave it for future study.

After writing the article, we found that Encyclopedia Talmudit, entry “Gezerah Shavah” (from note 143 onward), cites additional examples of such a chain. Indeed, it is brought there that Rashi on Babylonian Talmud, Ketubot 45b, and likewise in Halikhot Olam, Laws of Gezerah Shavah, wrote that this is an asmakhta, in complete agreement with what we independently concluded from his comments in the Eruvin discussion. It seems, however, that they did not notice that this is not a chain of gezerah shavah alone, but a chain of hekkesh and gezerah shavah in alternation. What misled them was probably Nahmanides’ expression in his glosses to the second root, where he calls such a structure “a gezerah shavah built upon a gezerah shavah” (see note 149 there). In this week’s article we shall expand on that point.

Difficulties at the end of the midrash

At the end of the Sechel Tov passage cited above there is an interesting, and perhaps even surprising, addition. The midrash sees fit here to define the essence of hekkesh and gezerah shavah, and it also states that neither may be expounded on one’s own:

“A hekkesh is where the matters are not similar to one another, neither in writing nor in pronunciation, but since both are written in one verse we learn one from the other—for example, ‘place’ from ‘flight,’ and ‘flight’ from ‘boundary,’ and ‘boundary’ from ‘outside.’ A gezerah shavah is two matters that resemble one another, and although they are not in one verse we learn one from the other—for example, ‘flight’ from ‘flight,’ and ‘boundary’ from ‘boundary,’ and ‘outside’ from ‘outside,’ and so throughout the Torah. And the Rabbis said: Kal va-homer is given to every person to expound; gezerah shavah and hekkesh were not given to be expounded, except where the Sages expounded them according to the tradition in their hands.”

First, it is not clear why the midrash finds it necessary, specifically here, to enter into general definitions of these principles. Second, why is it important, specifically here, to distinguish between hekkesh and gezerah shavah? Third, it is not clear why the author of the midrash found it necessary, specifically here, to note the rule that a person may not derive gezerah shavah and hekkesh on his own.

We should note that the language of the midrash indicates that, despite the difficulty of taking this passage as a real derivation—we saw above that according to Rashi it appears to be an asmakhta—it nonetheless regards these derivations as real. Therefore it learns from this very structure how one ought to relate to gezerah shavah and hekkesh, just as the Meiri above also learned principles concerning gezerah shavah from this derivation.

The midrash adds that unlike kal va-homer, which a person may derive on his own, hekkesh and gezerah shavah were derived only by the Sages according to the tradition in their hands. We have already mentioned several times that Rashi and Tosafot on Babylonian Talmud, Sukkah 31a, see Tosafot there, s.v. “Ve-Rabbi Yehudah savar,” disagree regarding the other principles: Tosafot hold that a person may indeed derive hekkesh on his own, whereas according to Rashi only kal va-homer may be derived independently. Our midrash apparently accords with Rashi.³

Perhaps precisely because the passage is so difficult, and yet this midrash apparently presents it as a full derivation, it finds it necessary to apologize and explain that the derivation was transmitted by tradition and was not invented for the occasion. According to our suggestion, Tosafot in Sukkah, who holds that a person may derive hekkesh on his own, is not refuted by this. It may be that with respect to such a complex hekkesh he too would require tradition.⁴

Defining the links of the chain: between hekkesh and gezerah shavah

The concluding passage of the midrash presents these two principles as duals of one another: gezerah shavah is based on similar words that appear in two different places. Hekkesh, by contrast, links two different words that appear in the same place.

One may say that there is here an interplay between two parameters: textual “distance” and verbal similarity. In gezerah shavah there is verbal proximity but textual distance. In hekkesh there is textual proximity, since the words are in the same verse, but verbal distance.

The general structure of the two-dimensional “midrashic network”

The midrashic chain used here is composed of two kinds of links: hekkesh and gezerah shavah. We saw that they possess dual properties. If we view the Bible as a space in which midrashic movement takes place, we have here a network composed of local links, that is, hekkesh, located within a single verse, and links with spatial spread, that is, gezerah shavah, which connect two different places.

At first glance, this seems to have the form of a tree, in the mathematical sense, composed of vertices and branches linking the vertices. The vertex is a point-connection, namely a hekkesh between two words in the same verse, that is, in the same location. The branch is a linear connection, namely a gezerah shavah between two different contexts, or locations. See Figure 1. But according to this description, this is a trivial tree, made up of only a single path with no branching at all: from each vertex only one branch exits and only one branch enters, apart from the two endpoint vertices. In such a case, the distinction between vertices and branches is apparently not important. In the language of hermeneutical principles, we may ask: why do we need two different kinds of links in such a structure at all? Why can we not make do with only one kind?

The answer is that the preceding description is incomplete. In fact, this is a two-dimensional structure, see Figure 2, whose two axes are verbal “distance,” that is, similarity and difference between words, and textual “distance,” that is, distance between verses in the Bible. We shall now propose two explanations for the two-dimensionality of this structure.

[Two figures of the graph of the derivation. Note on the order relation that defines an axis.]

Two explanations for the two-dimensionality of the structure

1. The aim of this structure is to connect the two endpoints of the tree, the Levitical cities and the Sabbath limits. These two points are distant from one another both verbally—there is no common word between the Levitical cities and the Sabbath limits, for otherwise we would make a direct gezerah shavah between them—and textually—the two topics are found in different and distant verses, for otherwise we would make a direct hekkesh between them. To connect such points, we must move along both the verbal axis and the textual axis. Therefore, the graph before us is a path in a two-dimensional space.

That is why two kinds of links are needed: one advances us along the verbal axis, namely hekkesh, and the other along the textual axis, namely gezerah shavah. The combination of the two can bridge a two-dimensional gap between two endpoints.

In this picture we no longer have two kinds of links, local and extended, but rather two kinds of extended connections: one along the verbal axis and the other along the axis of textual distance. The points connected by these links are two-dimensional points characterized by two coordinates: a particular word and a location in the text.

2. Let us look at this from a different, more general angle. Here we ignore the character of the gap between the endpoints we want to connect—that was the starting point of the first explanation—and examine a priori the structures that can arise with only one kind of link.

If we were to build a structure out of a series of hekkeshim alone, that structure could connect only concepts that appear in the same verse. But if concept A appears in the same verse as concept B, and concept C also appears in the same verse as B, and concept D in the same verse as C, then concept D likewise appears in the same verse as A. If so, we can make a direct hekkesh between them, and there is no need for the mediating hekkeshim. In such a case, the complex structure is superfluous.⁵ The same is true of gezerah shavah, except that here the argument concerns verbal distance rather than textual distance. A successive series of gezerot shavot is irrelevant, since all the words are similar to one another, including the words at the endpoints. Here too, one can always perform a direct gezerah shavah between the similar words at the two endpoints, without any need for the mediating links.

This consideration shows that a complex structure must, by definition, be built from two different kinds of links.⁶ Any single-type structure can be reduced to a single link.

Are there hermeneutical principles that are “diagonal” links?

We may now ask whether there is a single hermeneutical principle that can connect two endpoints that are distant in both dimensions. In other words, is there a hermeneutical principle that connects two subjects that are both far apart and expressed in different words?

One might perhaps view binyan av or kal va-homer in this way. The subjects that these principles connect are distant from one another both verbally and textually. But this is an illusion, for two reasons:

1. These principles are not textual. The considerations that connect the different contexts are conceptual rather than textual, and therefore these principles cannot be integrated into a graph made up of gezerah shavah and hekkesh.
2. In these two principles there is only one biblical vertex. The second vertex is created by the derivation itself. For example, when we learn by kal va-homer from liability for damage caused by an ox’s horn in the public domain to liability in the injured party’s courtyard, the liability in the public domain appears explicitly in Scripture. But liability in the injured party’s courtyard has no biblical source. Precisely for that reason a kal va-homer derivation is needed to teach that it too exists on the map of tort law. These principles therefore cannot function as links in a complex structure like the one in our midrash, which connects existing biblical points—word and location.⁷

We are therefore looking for textual hermeneutical principles that connect distant subjects described in different words.

It seems that there is no such principle in Rabbi Ishmael’s baraita of the principles. Perhaps the derivation of semukhin can be such a principle. In a semukhin derivation we compare subjects that appear in adjacent verses. A complex structure made of many links of adjacency can certainly create a derivation that cannot be created by a single link of semukhin. Unlike a chain of hekkesh-links, which cannot leave a single verse, a chain of semukhin-links can indeed create a structure with whatever breadth we wish, and it is also not committed to verbal similarity. To be sure, some dispute the derivation of semukhin, see for example Babylonian Talmud, Berakhot 21b and elsewhere, and it also does not appear in Rabbi Ishmael’s baraita.⁸

The relation between hekkesh and gezerah shavah

We have thus learned that gezerah shavah and hekkesh are two dual principles that spread across the two-dimensional space whose two axes are verbal distance and textual distance. As we have seen, at least within Rabbi Ishmael’s system one cannot move diagonally but only parallel to the axes, so that each advance is along only one axis. We also saw that only the combination of the two can create a complex derivation that cannot be performed by means of a single link.

That is to say, these two are the fundamental principles that are exclusively responsible for exposing distant correlations in the biblical text along those two axes. Every such connection can be unfolded by combinations of gezerah shavah and hekkesh alone.

What does this imply about the character of these principles? As we saw in the Sechel Tov passage above, both stand in contrast to kal va-homer. A person may derive kal va-homer on his own, but not these two.⁹ We know that additional rules stated about gezerah shavah are generally applied to hekkesh as well. For example, the rule that a hekkesh cannot be applied partially parallels the rule that a gezerah shavah cannot be applied partially.¹⁰ It therefore seems that these two principles share a similar character.

A note on the absence of hekkesh from Rabbi Ishmael’s baraita of the principles

The commentators ask why the principle of hekkesh is not counted in Rabbi Ishmael’s baraita. Some wrote that it is included under “a matter learned from its context or from its conclusion,” see Sefer Keritut, Principles of Hermeneutics, section 2, chapter 14, s.v. “Ve-od yitakhen lefaresh.” In Middot Aharon, part 3, section 6, it is explained that hekkesh is included under gezerah shavah, “for they are of one type, to compare two matters on the authority of the Giver of the Torah even though they are not equal in essence.” See Encyclopedia Talmudit, entry “Hekkesh,” note 19.¹¹

The necessity of similarity between these two principles

We have already raised several times the question: what is gezerah shavah? We saw that it can appear in two forms:

1. clarification of meaning, that is, illuminating the meaning of a biblical word or concept;
2. a textual means for transferring laws from one context to another.

We asked whether this device is merely technical, or whether it is a textual indication of a substantive similarity between the contexts being compared. The same question can be asked about hekkesh. But from our discussion here one conclusion follows: whatever the character of these two principles may be, it is similar.

The very combination of these two principles into one shared structure clearly indicates that they have a similar character. One cannot build a complex structure out of two links made of principles of different character. If gezerah shavah is a textual indication of the existence of a substantive similarity between the contexts, then hekkesh must be so as well. If gezerah shavah taught us similarity, while hekkesh were merely a technical tool that transfers laws or semantic meanings, then it would be impossible to combine them into one complex structure. Let us explain this a bit more.

Suppose that the gezerah shavah “place”-“place,” which is the first link in the midrashic sorites above, were a textual indication of similarity between the contexts involved, namely Sabbath limits and accidental homicide. After that comes the hekkesh “flight”-“place,” which links those words to one another; in this case both appear in the same context, that of accidental homicide and the blood-avenger. If the hekkesh were merely technical, then the word “flight” would bear no substantive similarity to the word “place,” that is, to Sabbath limits. Now comes the gezerah shavah “flight”-“flight,” which links two contexts to one another; in this case too, within the same section. But according to our assumption, gezerah shavah is an indication of similarity, and therefore the second “flight” is similar to the first “flight.” Yet the first “flight” is not similar to “place”; it is only linked to it by hekkesh. If so, there is no basis for finding similarity through a two-link structure between “flight” and “place,” since we are comparing things that are not similar.

The conclusion is that the character of gezerah shavah and hekkesh must be similar, for otherwise it would be impossible to compose them into one overall structure. Either both are indications of the existence of similarity, or both are technical tools for transferring legal information between biblical contexts.

Of course, all this is only according to the Meiri and Tosafot on Sotah, and according to the midrash Sechel Tov, who hold that this is a real derivation and not an asmakhta. According to Rashi, one may say that all this is indeed nothing more than an asmakhta, since he may not regard these links as real derivations at all. On his view, none of our conclusions is relevant.

“Learned from the learned”

As is well known, in the realm of sacrificial law there are restrictions on applying a hermeneutical principle to a law that is itself learned from a derivation. In other areas of Jewish law there is no such restriction. Is every combination of two principles possible, or must there be similarity between the principles compounded one upon another?

If we examine the Talmudic discussion of “learned from the learned,” we will see that the Talmud proposes all kinds of combinations. Some of them are impossible in sacrificial law and some are possible, but in ordinary law they are all possible. Therefore every combination mentioned there, even those that are not possible in sacrificial law, is apparently a logically possible combination.

The Talmud there connects hekkesh, which is a textual principle, with kal va-homer and binyan av, which are conceptual principles; see Encyclopedia Talmudit, entry “Hekkesh,” chapter 6. If so, it would apparently seem that one can indeed build structures out of two principles belonging to different types. Does that not contradict what we said above, namely, that one structure cannot be composed of principles of different character?

If we pay attention to the detailed argument presented in the section before the previous one, we will see that a logical problem in the structure arises only when one principle reappears twice. If we combine gezerah shavah with hekkesh on the assumption that their characters differ—namely, that gezerah shavah is an indication of substantive similarity, whereas hekkesh is a technical tool for transferring laws—then no problem arises. The problem arises only when we complete an overall gezerah shavah that contains within it two links of gezerah shavah and one of hekkesh. The problem is that we have lost the similarity between the contexts halfway through. But a two-link structure poses no problem.

For example, let us suppose that the gezerah shavah “place”-“place,” which is the first link in the midrashic sorites above, were a textual indication of similarity between the contexts involved, namely Sabbath limits and accidental homicide. Then comes the hekkesh “flight”-“place,” which links those words to one another; here both appear in the same context, accidental homicide and the blood-avenger. If the hekkesh were merely technical, then the word “flight” would bear no substantive similarity to the word “place,” that is, to Sabbath limits. But that, in itself, is not problematic, since the composite can now function as a technical device even though there is no substantive similarity between its two endpoints.¹² The problem arises only when we take the second endpoint and continue it by means of yet another gezerah shavah, since that next gezerah shavah again seeks to find substantive similarity, whereas the conceptual similarity has already been lost earlier along the way.

The logical sorites

As we have already noted, the Greeks called a structure of chained logical inferences, in which one is built upon another, a sorites. Let us take an example of a sorites in that sense and see the differences between it and the midrashic sorites that we encountered here.

In Greek logic there are sorites structures that connect two successive arguments into one comprehensive argument. Such a structure is needed when, from our premises, we cannot derive the conclusion we seek by means of any one of the logical rules at our disposal. In such a case, we build a schema composed of two logical rules, which may also be identical rules, in which the conclusion of the first argument serves as a premise in the second. In this way one can derive the conclusion of the second argument from the premises of the first.

In such a structure there is no substantive novelty beyond the basic structures of which it is composed. The composition is nothing more than a combination possessing the same validity as its components. All of them are deductive rules, and therefore necessary. The composite whole can itself function as an independent rule in our logic, although usually, for the sake of theoretical economy, it is presented as a composition of the two more basic rules.

Aviram Ravitzky, in his edition of Sha’arei Tzedek, writes that the author presents the derivation of the “common denominator” as a kind of sorites; see his introduction to that edition, pp. 14-15. Another picture of a sorites may be seen in the Zevahim discussions of “learned from the learned,” where one hermeneutical principle is compounded upon another. There too we derive the conclusion of the second derivation from the premises of the first through a complex midrashic structure. The case with which we dealt here is likewise a structure that combines hekkesh with gezerah shavah, except that it is “longer,” that is, it contains more compound links: a composition of compositions.

A midrashic sorites

Here we should note an important difference between the logical sorites and this structure, a difference that becomes more pronounced as the chain grows longer. There is a fundamental difference between deductive logical inference and analogical inference. Deductive inference possesses full validity, and therefore extending the chain so that it contains several deductive links does not alter the degree of validity of the overall argument. The entire logical chain, however long, has full validity. Analogical inference, by contrast, rests on some degree of similarity between the sides being compared. By its very nature, the longer the chain becomes, the weaker the similarity between the endpoints.

An example is the derivation in Menahot: the blue thread resembles the sea, and the sea resembles the sky, and the sky resembles the Throne of Glory.

This is the kind of argument that can ground the problem the Sages see in derivations based on chains of “learned from the learned.” Even so, as we have already mentioned several times, outside the domain of sacrificial law there is no obstacle to deriving in a structure of “learned from the learned.”

Our midrash is built as a very long chain. If we assume that each link expresses some similarity between its two ends, then it is reasonable to think that the degree of similarity steadily diminishes as the chain grows longer. The degree of similarity that remains in a very long chain is already rather limited, and it becomes very difficult to build on that basis an analogical inference between the endpoints.

This consideration supports Rashi’s position, which sees this midrash as an asmakhta rather than as a full derivation. It is difficult to see substantive similarity surviving after such a great “inferential length.” Let us recall that Rashi’s understanding of the rule that a gezerah shavah cannot be partial is that gezerah shavah is an expression of substantive similarity. Therefore, unlike Rashbam, he explains “not partially” to mean that the derivation must work in both directions. See the supplement to the sheet for Parashat Vayera 5765, and elsewhere.

On the other hand, nowhere in rabbinic literature have we found a limitation on the length of derivations, or on compounding hermeneutical principles one upon another. That seemingly indicates that there is no obstacle to combining long chains of derivations. In practice, however, despite the absence of a formal limitation, we find only very few cases in rabbinic literature of derivations composed of chains at all, and certainly chains of more than two links, even in legal contexts that do not belong to sacrificial law.¹³

Two additional directions for validating the midrashic sorites

We saw that from the analogical character of the hermeneutical tools there follows, almost inevitably, the conclusion that the validity of a midrashic inference diminishes as the chain grows longer. This problem can be solved, even if we continue to hold that sorites-derivations are not asmakhta, in two ways:

1. If we adopt the assumption that these derivations function as a semantic conduit, and not as an expression of substantive similarity. According to this view, gezerah shavah and hekkesh are technical tools for transferring legal information, not expressions of substantive similarity. When there are two similar words, or two concepts in the same verse, that is a textual cue that we are to transfer laws from one context to the other. In this way one can continue to regard the chain as a full derivation and not as an asmakhta, while still claiming that its validity does not decline. This would explain the fact that we do not find in rabbinic literature a limitation on the length of the midrashic chain.¹⁴

2. According to the conception of analogy as revelatory.¹⁵ As we saw in the sheets for Parashat Lekh Lekha and Parashat Yitro, 5765, sometimes analogical inference is indeed a tool based on similarity and comparison, but it may nevertheless reveal a logical identity. We brought an example from the discovery of scientific laws. The method used by scientists is inductive, that is, a generalization from particular cases, but our confidence in the law revealed in that way is very high. The reason is that the mode of discovery does not always indicate the character of the law discovered. Scientific particulars, such as the fall of several bodies to the earth, provide the basis for an inductive generalization and the inference of a general law that all bodies are attracted to the earth. Once the generalization has been made, we adopt that law as binding for all bodies with mass. Some would say that this law is analytically derived from the essential properties of the concept of mass.¹⁶

According to this conception, one may say that although each link in the chain has an analogical character, it reveals a substantive logical relation between the two sides. For example, the gezerah shavah reveals that the two sides ought to be logically identical to one another from a legal point of view. Despite the analogical character of the hermeneutical principles that we use, the legal conclusion that emerges from them does not have the force of analogy but of deduction, as in the example of scientific laws above. If so, then even a long chain expresses a necessary logical relation between the endpoints, and the force of that relation does not fade with the length of the chain, much like the logical sorites.

Footnotes