Korach (5764)

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 — Friday eve, Parashat Korach 5765

Questions

1. What do slaves, documents, and land have in common?
2. Defining concepts by extension and by intension.
3. What is the connection between De Morgan’s laws in logic and the rule of generalization, specification, and generalization?
4. Does the school of Rabbi Yishmael depart from its usual practice and sometimes interpret by inclusion and exclusion?
5. Do all the rules of generalization and specification belong to a single form of reasoning?
6. What does one do when Scripture presents a non-canonical pattern, such as specification-specification-generalization?

The Hermeneutical Principles

• Generalization and specification.
• Specification and generalization.
• Generalization, specification, and generalization.
• Inclusion, exclusion, and inclusion.
• The common denominator.

“Every first issue of the womb of all flesh that they offer to the Lord, of man and beast, shall be yours; but you shall surely redeem the firstborn of man, and the firstborn of an unclean beast you shall redeem. And those redeemed from it, from a month old, you shall redeem, according to your valuation, five shekels of silver by the sanctuary shekel, which is twenty gerahs.”

— Numbers 18:15-16

Mishnah: One may not redeem with slaves, nor with documents, nor with land, nor with consecrated property…

Gemara: The Mishnah is not in accordance with Rabbi, for it was taught in a tannaitic teaching: Rabbi says, a firstborn son may be redeemed with anything, except documents. What is Rabbi’s reason? He interprets by inclusion and exclusion: “And those redeemed from it, from a month old, you shall redeem” — an inclusion; “according to your valuation, five shekels of silver” — an exclusion; “you shall redeem” — an inclusion. Inclusion, exclusion, and inclusion — this includes everything. What does it include? It includes all things. And what does it exclude? It excludes documents.

The Rabbis interpret by generalization and specification: “And those redeemed from it, from a month old” — a generalization; “according to your valuation, five shekels of silver” — a specification; “you shall redeem” — it again generalized. In a case of generalization, specification, and generalization, you judge only by what is like the specification. Just as the specification is explicitly something movable and itself a form of property, so too everything movable and itself a form of property. Land is excluded, because it is not movable; slaves are excluded, because they are equated with land; documents are excluded, because although they are movable, they are not themselves property.

Ravina said to Mareimar: Does Rabbi really interpret by inclusion and exclusion? But Rabbi interprets by generalization and specification in the case of the awl… And we say: What is the point of dispute there? Rabbi interprets by generalization and specification, while Rabbi Yosei son of Rabbi Yehuda interprets by inclusion and exclusion. Yes: generally Rabbi interprets by generalization and specification, but here he follows the school of Rabbi Yishmael, for the school of Rabbi Yishmael taught: “in the water, in the water” twice, in Leviticus 11 — this is not generalization and specification but rather inclusion and exclusion. And the Rabbis say, as they say in the West: Wherever you find two adjacent generalizations, insert a specification between them and judge them as generalization and specification.

— Babylonian Talmud, Bekhorot 51a

A. Redemption of the Firstborn Son

Introduction

In our weekly Torah portion we find the commandment to redeem the firstborn son; see also the essay on Parashat Ki Tissa. A human firstborn is redeemed from the priest for five selaim, either in actual money or in some item of equivalent monetary value.

The Mishnah in Bekhorot quoted above states that one may not redeem a firstborn son by means of slaves, documents, or land. These are three exceptional types of monetary equivalent: if one redeemed his firstborn son with them, the redemption is invalid. The Gemara explains that the source of this rule is a derivation by the hermeneutical rule of generalization, specification, and generalization, which broadens the concept of “money.” One may indeed redeem with a monetary equivalent and not only with coinage, but the equivalent used for redemption must resemble money in some sense. It must be movable and itself a form of property, like money, and therefore documents, slaves, and land are invalid for redemption.

That same Talmudic discussion also presents a tannaitic dispute, between Rabbi Yishmael and Rabbi Akiva, over whether one interprets by inclusion and exclusion or by generalization and specification, with consequences for the halakha (Jewish law) of redeeming the firstborn son. We have already encountered this dispute in the essay on Parashat Toldot, at the end of volume I, in the essay on Parashat Va’era, and in greater detail in the essay on Parashat Vayikra. In the Talmudic discussion in Bekhorot quoted above, however, there is a certain complication, since Rabbi adopts a position that in some circumstances combines the methods of Rabbi Yishmael and Rabbi Akiva. It should be noted that the Gemara says that Rabbi Yishmael himself had already done something similar. Before discussing that novelty, let us briefly return to what we have already seen about this dispute.

Brief Summary: Generalization and Specification, Generalization, Specification, and Generalization, and Inclusion, Exclusion, and Inclusion

As noted, the rule with which our discussion is concerned is generalization, specification, and generalization. This is one of the family of rules of generalization and specification. The mode of operation of this rule was described in the essays on Parashat Va’era and Parashat Vayikra. There we showed that the use of generalization, specification, and generalization is nothing more than an expanded application of the principles used in the rules of generalization and specification and specification and generalization.

In the essay on Parashat Vayikra, volume III, we pointed out that the rule of generalization, specification, and generalization turns toward cases that bear similarity in one aspect only, one parameter, between the teaching specification and the set of cases derived from it — in Rabbi Yishmael’s terminology, “like the specification.” This is a limited similarity, which allows several parameters — usually two additional “sides,” in talmudic terminology — that characterize the teaching specification but are not found in the derived cases. This possibility naturally expands the group of derived cases. By contrast, the rule of generalization and specification yields derived cases that are fully similar to the teaching specification, in all three “sides,” in talmudic terms, thereby narrowing the scope of that group. In Rabbi Yishmael’s terminology this is called: “nothing but what is in the specification.”

As noted, in a derivation of generalization, specification, and generalization, the set of derived cases is broader than the set created by the rule of generalization and specification. As we already remarked in those essays, the smaller the number of characteristics that define a given set, the larger the number of items included in it, that is, the number of items characterized by those characteristics.

By contrast, interpretation by inclusion and exclusion turns the entire picture upside down. According to Rabbi Akiva, who interprets by inclusion and exclusion, a biblical pattern of generalization, specification, and generalization is understood as an almost total expansion of the specification, that is, “it includes everything.” There is only one exceptional case on account of which the biblical pattern appears.¹ As the early commentators explain, were it not for that exceptional case, the whole structure of generalization, specification, and generalization would be superfluous, for we would include everything from the specification by means of a binyan av, that is, a derivation from a paradigmatic case.

The Tannaitic Dispute: Generalization, Specification, and Generalization or Inclusion, Exclusion, and Inclusion?

The Gemara explains that these expositions are based on a threefold biblical pattern of generalization-specification-generalization. The first general clause is “And those redeemed from it, from a month old.” The specification is “five shekels of silver.” And the final general clause is “you shall redeem.” As Rashi explains there, the final general clause in fact appears before the specification. This is a central point in the discussion, and the Gemara addresses it later on.

The Gemara explains that the tanna of the Mishnah disagrees with Rabbi, whose opinion appears in the baraita, about the interpretive method to be applied to this biblical pattern. According to the tanna of the Mishnah, one interprets by generalizations and specifications; according to Rabbi, by inclusions and exclusions. As we have already seen, one who interprets by generalization, specification, and generalization arrives at “what is like the specification.” Therefore, according to the tanna of the Mishnah, redemption is possible only with things that resemble the specification. This is an expansion of the specification — five silver selaim — into a class that includes all movable items whose substance is itself property; these are the relevant aspects of resemblance. Hence one may not redeem with something that is not movable and whose substance is not itself property, namely slaves, documents, and land.

By contrast, according to Rabbi, who interprets by inclusion, exclusion, and inclusion, the result is that the specification expands to include all things, that is, anything of monetary value. There is indeed one element excluded from this expansion even according to Rabbi — as noted in the summary above — namely documents. According to Rabbi too, one may not redeem with documents.

What characterizes documents, which Rabbi chose to exclude? At first glance, he excludes one of the axes employed by the tanna of the Mishnah. He includes even something that is not movable, namely slaves and land; however, he is not willing to permit redemption by means of something whose substance is not itself property, namely documents. In the essay on Parashat Vayikra we pointed out that sometimes inclusion, exclusion, and inclusion excludes a detail that is unrelated to the general axes of similarity — in that case the excluded item does not resemble the written specification in any aspect at all — and sometimes what is excluded is one of those axes themselves. In that latter case, one is effectively expanding the specification only along one of the parameters of similarity, whereas Rabbi Yishmael learns from the specification to a broader group by expanding it along all of the axes of similarity.

The Course of the Sugya

Later in the discussion, the Gemara raises a contradiction within Rabbi’s own view. On the one hand, here in the baraita we have seen that he interprets by inclusion, exclusion, and inclusion. On the other hand, in the discussion in Babylonian Talmud, Kiddushin 21a, there is a tannaitic dispute about how to interpret the verses dealing with the boring of a slave’s ear. There it turns out that Rabbi specifically interprets by the rule of generalization, specification, and generalization, contrary to the method he uses here, while Rabbi Yosei son of Rabbi Yehuda, who disagrees with him there, is the one who interprets by inclusion, exclusion, and inclusion.

One should notice the assumption underlying the Gemara: there must be consistency in the use of these hermeneutical rules. One who interprets by generalization and specification must always do so, and not by inclusion and exclusion. And one who interprets by inclusion and exclusion must always use those rules, and cannot use the rules of generalization and specification. The same picture appears in Babylonian Talmud, Shevuot 26a, which we already discussed in the essay on Parashat Vayikra. There too the dispute is presented as a comprehensive methodological dispute, not as a disagreement about the interpretation of one particular set of verses. These are two schools of study, and each sage must belong to one school alone.

As we have already remarked several times, in the Amoraic period this dichotomy begins to blur. In the Gemara one can see a number of mixtures of the interpretive methods of Rabbi Yishmael and Rabbi Akiva. We saw that even in the legal rulings of Maimonides there seems, at least at first sight, to be no consistency with respect to interpretive method. But all of this occurs specifically in the Amoraic period. In the Tannaitic period, and even in the generation of Rabbi — who was among the last, or next-to-last, generation of tannaim — the boundary between the two schools was still carefully maintained.

Yet in the Gemara’s resolution there already seems to be some retreat from that strict division. The Gemara explains that Rabbi in fact belonged to the school of Rabbi Yishmael, but that here this is an exceptional case in which even he interprets by inclusion and exclusion rather than by generalization and specification. The reason is that in our biblical pattern the final general clause appears before the specification. In such a case, even the school of Rabbi Yishmael interprets by inclusion and exclusion rather than by generalization and specification.

The Gemara nevertheless goes out of its way to bring a source for this from a teaching of the school of Rabbi Yishmael concerning water-creatures, in Babylonian Talmud, Hullin 67a,² where we also find such a pattern, and there too the school of Rabbi Yishmael interpreted by inclusion and exclusion. That is, the division between the two schools is preserved, even if its content and sharpness are somewhat softened. In certain cases, even the school of Rabbi Yishmael is willing to adopt the method of inclusion and exclusion, and does not use only the rules of generalization and specification.³

The Meaning of This Qualification

Our sugya presents a qualification on the use of the rules of generalization and specification. At first glance, this qualification is a technical-formal rule, which states that in the ordinary case in which Scripture presents a pattern of generalization-specification-generalization, we are to generalize the specification to the level of similarity in one respect — and that is the rule of generalization, specification, and generalization. But when Scripture presents a pattern of generalization-generalization-specification, we are to generalize to the entire field, with the exception of one anomalous case. This qualification does not seem to possess any internal logic. It appears to be simply part of the tradition given to Moses at Sinai concerning the use of the hermeneutical rules.

Yet it seems that this distinction can still be anchored in the general textual conception behind the rules of generalization and specification; see the essays on those Torah portions. We have already seen several times that a pattern of generalization and specification means that the specification defines the group, out of the broad generalization, to which the relevant halakha applies. Therefore, in such a pattern the group of derived cases is narrow: “nothing but what is in the specification.” When another generalization is added at the end of the structure, the Torah instructs us to expand this narrow group into a broader group of items similar to the specification in one aspect: “only what is like the specification.”

What happens in a case like ours — generalization-generalization-specification — in which the final general clause appears before the specification? In such a case there is no narrowing of the first general clause, since no specification appears after it. The first general clause therefore remains in force, and the halakha applies to the whole range of cases. Yet one cannot ignore the fact that after the first general clause there appears an additional structure of generalization and specification. The Gemara says that in such a case we conclude that the Torah nevertheless wishes to exclude something — but something defined and narrow. How should we understand that?

A Proposal That Places This Derivation Within the Framework of Generalization and Specification

It is possible that the structure of generalization and specification that appears after the first general clause is itself interpreted like an ordinary case of generalization and specification, that is, as applying to a narrow group defined as “nothing but what is in the specification.” The difference is that in such a biblical pattern this narrowing and limitation is not stated with regard to the process of expansion, as in the ordinary case of generalization and specification, but rather with regard to the qualification of the first general clause. This is a narrowing of the qualification, which in the end creates an expansion.

In such a pattern, the specification does not define the first general clause, but rather the final general clause. In other words, the specification tells us that the expansion of the first general clause is not sweeping; it has one narrow qualification. The phrase “nothing but what is in the specification” marks not the domain in which the relevant halakha applies, but the domain to which it does not apply. The halakha derived applies to the entire field except for one narrow case — “nothing but what is in the specification.” This is exactly the result of a derivation by inclusion, exclusion, and inclusion. As we have seen, in that hermeneutical rule the relevant halakha applies to the whole field except for one narrow case.

If this proposal is correct, then the interpretation of a biblical pattern of generalization-generalization-specification does not really entail a substantive move into the school of Rabbi Akiva. The result is indeed parallel to a derivation by inclusion, exclusion, and inclusion, but we arrive at that result through a form of exposition that belongs entirely to the rules of generalization and specification of the school of Rabbi Yishmael. The school of Rabbi Yishmael, including Rabbi within it, remains methodologically consistent within the domain of generalization and specification; only in this particular case does the result happen to be equivalent to a derivation by inclusion, exclusion, and inclusion.⁴

A Linguistic Objection to Our Proposal

There is room to object to our proposal on the basis of the Gemara’s wording. The Gemara says that we take the specification and place it between the two generalizations, and then interpret by the rule of inclusion, exclusion, and inclusion. The simple meaning of that language is that we are really using Rabbi Akiva’s hermeneutical rule. According to our proposal, however, the result is produced specifically by the fact that the specification appears after the two generalizations. If we were to place it between them, then apparently we ought to return to a derivation by generalization, specification, and generalization.

According to our proposal, there is no escaping the conclusion that this wording is only a shorthand formula, meaning roughly this: treat such a biblical pattern as though one had inserted the specification between the two generalizations and then interpreted by inclusion, exclusion, and inclusion. Even if that is not what we are actually doing, the result yielded by that process would be the correct one.

The View of the Sages

The sugya concludes by explaining the view of the Sages, that is, the tanna of the Mishnah, who disagree with Rabbi. The Gemara determines that both they and Rabbi belong to the school of Rabbi Yishmael, that is, to those who interpret by means of the rules of generalization and specification. The dispute between them concerns what we are to do in a pattern of generalization-generalization-specification. In Rabbi’s view, we insert the specification between the two generalizations and interpret by inclusion, exclusion, and inclusion. By contrast, the Sages adopt the position of the sages of the Land of Israel: even if the final general clause appears before the specification, we still place the specification between them, but we interpret the resulting structure by generalization, specification, and generalization. In our terms above, we may say that for them the specification at the end defines the first general clause, and the final general clause then broadens this again into the class defined as “what is like the specification.”

Why, in fact, do the Sages not interpret the pattern of generalization-generalization-specification using the ordinary logic of generalization and specification? In other words, why do they not take account of the real position of the general clauses and the specification, but instead move them from their actual position to a more familiar position in order to interpret them? It seems that there is here a different conception of the hermeneutics of generalization and specification.

From the words of the early commentators it emerges that patterns of generalization and specification are interpreted consistently in the same way. Whenever the specification appears after the generalization, it defines and limits it to the group of “nothing but what is in the specification.” Whenever a generalization appears after a specification, it broadens it in one aspect of similarity. These two rules can be applied in whatever form or structure generalizations and specifications appear in Scripture. Admittedly, in the end every such form would produce a different result. From such a conception it follows that every biblical pattern that contains generalizations and specifications can be interpreted differently according to its structure. In fact, there should not be only three rules of generalization and specification, but infinitely many such rules.

By contrast, from the approach of the Sages and the sages of the Land of Israel here it emerges clearly that the structures of generalization and specification, specification and generalization, or generalization, specification, and generalization are not interpreted as particular cases of one unified scheme, or as parts of one general form of reasoning. It seems that according to the Sages there are only three specific rules of generalization and specification: generalization and specification, specification and generalization, and generalization, specification, and generalization. All other patterns are accidental, and either cannot be interpreted at all, or must somehow be treated by means of one of those three rules — for example, by changing the location of the specification for purposes of interpretation, as in our case.

Of course, once this is said, we must ask ourselves, according to the Sages: why did the Torah itself not place the specification between the two generalizations, and instead choose to write the verse in a non-canonical form? Considerations of plain-sense exegesis could sometimes provide an explanation, for at times the syntax of the sentence does not allow otherwise. But in our case, for example, that is not so. The matter therefore remains unresolved.

A Possible Implication

A possible implication of the picture described here would concern patterns such as specification-generalization-generalization, or generalization-specification-specification. According to our approach in explaining Rabbi’s view, it would be possible to apply to them the ordinary form of reasoning of the rules of generalization and specification, although there are several possible ways of doing so, and this is not the place to elaborate. In the second example, for instance, one could say that the first part is a case of generalization and specification, and it gives us “nothing but what is in the specification.” Then the final specification comes and tells us to qualify the result of the previous structure even further, reducing it to the specification alone — that is, not everything that falls under the specification, but this specific specification itself, and nothing more.

To decide in what way such structures, and others like them, should be interpreted, we need a better understanding of the form of reasoning underlying the three more familiar rules of generalization and specification. This is not the place to pursue that matter.⁵

B. Extension and Intension of Sets

Problems in the Rule of Generalization, Specification, and Generalization in the Context of Firstborn Redemption

Let us now return to the derivation by generalization, specification, and generalization in our Torah portion. From the biblical pattern of generalization, specification, and generalization we conclude that a firstborn son may be redeemed with any item of equivalent value that is movable and itself a form of property. From this we exclude documents, because they are not themselves property — they are merely evidence of a monetary claim and are not themselves monetary value; and land, because it is not movable; and slaves, because they are equated with land.

The question is why we chose precisely these axes of similarity. We could, for example, have chosen another axis of resemblance, such as any object made of silver, like the silver selaim in the specification, or any object whose number is five items, and so forth.

A further question is whether there is something common to the two features: movable, and itself a form of property. Are these two unrelated exclusions, or is there something shared behind them, and is that what is excluded from the redemption of the firstborn?

Another question is why slaves, which are equated with land, are excluded as well. Taken in themselves, slaves are both movable and themselves a form of property. They are merely compared to land, which itself is not movable. If so, why should they too be excluded? It seems from this that what disqualifies land is not the fact that it is not movable, but something else, and that same thing is also present in slaves. It is not clear what that feature is.

We can now ask all of these questions in a broader and more general way: is there something common to slaves, documents, and land, and is that what is excluded by this derivation? Is this form of derivation merely formal, or is there some substantive and comprehensive idea behind it, shared by the three excluded categories? If there is such a common element, it characterizes both slaves and land, and may therefore also teach us the meaning of the scriptural equation between them. It could then teach us the idea that lies behind the exclusion of slaves, documents, and land from the redemption of the firstborn.

An A Priori Consideration: There Must Be a Common Denominator

In the essay on Parashat Shemot we discussed the rule of binyan av from two verses; see also the essay on Parashat Tetzaveh. The logic underlying it is “the common denominator,” or “what do these have in common?” We saw that this logic raises the difficulty known as the objection from the stricter side, and that difficulty arises from the very schematic structure of the rule. The claim there was that this hermeneutical rule tacitly assumes that one halakha cannot have two independent causes. If the halakha applies to case A and also to case B, then the reason for its application must be some common factor that characterizes both cases — the common denominator. We will not elaborate here, and the interested reader is referred to that essay.

From this consideration it follows that even when we exclude from a derivation of generalization, specification, and generalization along two axes — movable, and itself a form of property — there must be some shared element present in both, and that is what is excluded. One may formulate this as follows: from these two characteristics we can infer, through the logic of the common denominator, the rest of the items of equivalent value with which one may not redeem. If so, the group of items of equivalent value with which redemption is impossible is characterized by some factor common to these two criteria: movable, and itself a form of property.

A Seeming Proof from the Derivation Itself

In the previous section we assumed that the derivation of generalization, specification, and generalization excludes two kinds of items with which one may not redeem firstborn sons. But that is not quite accurate. From the derivation of generalization, specification, and generalization we do not directly derive the exclusion of what is not movable and what is not itself property. The situation is the reverse: the verse mentions money, and we compare other items to it and include from it everything that is movable and itself a form of property, like money. This derivation is an inclusion, not an exclusion: any item of equivalent value that is movable and itself a form of property may be used for firstborn redemption. The exclusion follows only by implication: whatever is not movable, or is not itself a form of property, cannot be used for firstborn redemption. How, then, should we understand these two characteristics?

We already noted above that in a derivation of generalization, specification, and generalization, one includes everything that resembles the specification in one aspect. If so, it apparently follows that “movable” and “itself a form of property” together count as only one aspect of similarity. That in itself is proof that there is something common to these two properties, and that is what is included in the redemption of the firstborn.

There are two possible ways to understand this claim:

1. Behind the two properties there stands a third, more fundamental property, of which the first two are merely different expressions. This is the root of the logic that underlies the rule of the common denominator.
2. The two properties together create a third, more comprehensive essence, composed of both: being something that is movable and itself a form of property.

The first possibility describes the situation presented in the previous section. If the derivation were an exclusion, and two kinds of things were excluded, then we would have to search for what is common to the opposites of those properties, since every halakha has only one root. According to that approach, it is clear that there must be something common to “not movable” and “not itself property.” But if the derivation is an inclusion, then what is included is what is characterized by the new whole created by the two properties together: movable and itself property. That is precisely the second possibility above.

The Relation Between the Two Possibilities

The relation between these two possibilities is one of negation. If we view the derivation as an inclusion, then what is included is everything movable and itself a form of property. The consideration we drew from the language of the Gemara — that this counts as similarity in only one aspect — teaches us that this conjunction creates a third concept composed of the two characteristics. But if we look instead at the excluding side of the derivation, then clearly what is excluded is everything that is not movable or is not itself a form of property. In that case, the logic of the common denominator teaches us that there is a shared basis underlying those two characteristics, and each of them is only a particular example of it.

These two considerations are mirror images of one another. Each explains the other. The logical consideration says that if two things are excluded, there must be a common basis between them, and that is what was excluded. By contrast, if we look at what was included, then the claim is that what was included is something composed of these two characteristics, and it resembles the specification in one aspect, not in two independent aspects. In the language of the Rogatchover, one could say that these two characteristics are fused into a “chemical combination,” that is, a true compound, and not merely a “neighborly combination,” that is, a mixture.

From simple logic it follows that every inclusion defined by two different characteristics can be reformulated as the exclusion of one property consisting of the combination of their negations. In the same way, every exclusion of two different things can be reformulated as the inclusion of one thing consisting of the combination of their negations. This is the relation between possibilities A and B above. In fact, these are the logical principles known as De Morgan’s laws, which deal with the negation of conjunctions and disjunctions.

Methodological Note: The Complement Set

This is a systematic logical path that may help us uncover the common basis of two halakhic situations. Sometimes it is useful to examine the dual set — in the language of set theory, the complement set — which contains the union of the negations of those situations, and to search there for the common basis, in the second sense of “sharing”: fusion, or common foundation.

When we ask what is common to what is not movable and what is not itself property, we may find ourselves perplexed. These look like two different criteria with no evident connection. But if we look at the complement set — that which is movable and itself a form of property — we see that it contains everything that resembles money: movable and itself property. There is no single common root to the two characteristics “not movable” or “not itself property”; rather, there is a fusion between their opposites — both movable and itself property — into one compound.

If so, what these two have in common is resemblance to money. This is the biblical specification, and the whole derivation proceeds from resemblance to it. It is admittedly hard to offer a verbal definition of the connection between these two characteristics, but it is clear that something that is both movable and itself a form of property is a kind of monetary equivalent that resembles money very closely in its essential qualities.

Thus there is indeed no one shared root to which each of these properties is merely an expression or example. Yet there is a foundation created by their combination, by their fusion: a marketable object that has monetary value. This is a fitting substitute for money — an item of value that is truly like money.

Definition by Extension and by Intension⁶

How are we to define this fusion? In logic one distinguishes between two ways of defining a concept: definition by extension and definition by intension. A definition by extension defines a concept by means of the set of objects that fall within its scope. For example, the concept “democratic state” may be defined extensionally by pointing to all the states known to us as democratic. That is the extension set of the concept “democratic state.”

By contrast, one can also define the concept by pointing to the full set of its characteristics. According to this, a democratic state is a state governed by representatives chosen in free elections every few years, in which civil and human rights are protected, where there is separation of powers, and so forth. This is a definition through the characteristics of the concept, not through the objects included in its extension.

Let us now apply this distinction to our case. The definition “everything that is movable and itself a form of property” is the intensional definition of the kinds of entities with which one may redeem firstborn sons. By contrast, “every item of monetary value except slaves, documents, and land” points to the extension set of the rule — that is, to the concrete objects with which redemption may be performed. As we already mentioned above, the richer the intensional definition in properties, the smaller the number of entities in the extension set. Every property added to the intensional definition removes entities from the extension set, and vice versa.

Which of the Two Definitions Is More Fundamental?

There are analytic approaches that regard definition by extension as the more fundamental definition of a concept. The obvious criticism of such approaches is this: how do we know which states deserve the title “democratic”? Clearly, before pointing to the extension set, there is already some intuition regarding the intensional meaning of the concept. We filter the democratic states by measuring them against some content-based criterion, and thus construct the extension set. That is the intensional definition of the concept. If so, the intensional definition precedes the extensional one.

What nevertheless leads some thinkers to say that the extensional definition is more fundamental is the fact that the intensional definition cannot always be expressed in simple verbal form. This is true especially when the intensional definition includes a combination of several different properties among which no visible connection appears. For example, a democratic state is characterized by free elections, freedom of speech, civil and human rights, separation of powers, and so forth. It is quite difficult to ground all these properties in a single common basis. They seem like an accidental bundle of unrelated traits.

On the other hand, there is a sense that this whole does create an entity with a meaning of its own, beyond the mere conjunction of traits. After all, we do not define a concept such as “dust-ball” — everything made of dust and round. The reason is that this bundle has no significance beyond the conjunction of its traits. By contrast, the collection of properties that characterizes a democratic state does create a description with an overall significance. That is why this collection receives a special name of its own: “democratic state.” The combination of the characteristics of a democratic state is not based on one root, but the whole created by their fusion certainly does have meaning. It is not for nothing that all these characteristics were gathered under the wings of one concept.

Very often this meaning cannot itself be described verbally. We therefore attach to it a word in the language as a name, and then add the collection of traits in order to define that word. In such cases, we may sometimes prefer extensional definitions, because the hearer may grasp the overall idea we are aiming at specifically through examples better than through a list of typical characteristics.

Back to Our Earlier Questions

What emerges from our discussion is that there is a close relation between definitions by extension and by intension. The intensional definition characterizes what is included in the extension set, and vice versa. Above, we sought the common element in the extension set of the items of equivalent value that are not valid for redeeming firstborn sons: land, documents, and slaves. The answer apparently lies in the degree to which each of these resembles money. All of them, despite being items of monetary value, are not sufficiently similar to money. This is probably also the basis of the comparison between slaves and land, and that is why slaves too cannot be used for redeeming firstborn sons, although this is not the place to elaborate.

For this reason we can find additional halakhic contexts in which it is precisely slaves, documents, and land that are excluded; see, for example, Babylonian Talmud, Shevuot 37b, and parallel passages. It would appear that the common basis is not rooted specifically in the halakhic features of redeeming firstborn sons, but in the degree of resemblance among these items themselves. In all such contexts, what is required is an item of monetary value that resembles money very closely, beyond the mere fact that it has monetary worth, a feature shared by every monetary equivalent.

Footnotes