Inset image: Wikimedia Commons
Inset image: Wikimedia Commons
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If A, Then B

Through Daf Yomi’s exercises in mathematical logic, Talmudic rabbis attempt to decipher the will of a reasonable God. Plus: What distinguishes guilt from sin?

Adam Kirsch
May 01, 2018
Inset image: Wikimedia Commons
Inset image: Wikimedia Commons

Literary critic Adam Kirsch is reading a page of Talmud a day, along with Jews around the world.

At one point in this week’s Daf Yomi reading, Rabbi Yehoshua put a stop to the Gemara’s discussion by saying that it was halila, “going around in circles.” I often had the same feeling as we delved further into Tractate Zevachim, the section of the Talmud that deals with slaughtered offerings in the ancient Temple. That is because, while the subject matter of this tractate may seem highly concrete—it is all about killing animals and how to sacrifice their blood and flesh—it actually involves the kind of abstract logical reasoning you might find in a logic puzzle or LSAT question, which has never been my strong suit.

In general, I’ve found that different areas of the Talmud demand the exercise of different mental powers. Understanding the law of eruvs requires a geometrical imagination, while the law of permitted marriages requires a good memory for relationships. In Zevachim, what’s needed so far is the ability to parse fine logical distinctions—to take two things and determine how they are similar and how they are different. All of these abilities are related to logic and mathematics, and a rigorous Talmudic education stimulates those faculties—which may explain why so many great mathematicians have been Jewish.

Take, for instance, the problem that occupies the rabbis starting in Zevachim 10a. They have already established after lengthy discussion that a sin offering is disqualified if it is sacrificed “not for its own sake”—that is, if the priest offering it intended it for another kind of sacrifice. How do we know, the rabbis ask, that the same principle holds true of a guilt offering? (The basis of the distinction between sin and guilt offerings is a complex question in itself, which the rabbis do not address here.)

To answer this question, the rabbis have to find points of similarity between the sin offering and the guilt offering, which would justify their being treated similarly with respect to intention. Conversely, points of difference between the two kinds of offering tend to support the opposite conclusion, that a guilt offering is not disqualified if offered “not for its own sake.” This is the subject of a debate between RabbI Eliezer, who provides the similarities, and Rabbi Yehoshua, who brings up the differences. To clarify the argument (I hope), I will distinguish them by assigning each condition a letter.

The point that the discussion aims to settle can be called A: a sacrifice is invalid if sacrificed not for its own sake. We know that A holds true of sin offerings; how do we know whether it is also true of guilt offerings? Rabbi Eliezer begins with the most obvious point of resemblance: both sin and guilt offerings are “brought for a transgression.” Call this condition B, which both forms of sacrifice share: they are atonements for wrongdoing. Eliezer argues, to use the terms of symbolic logic, that if B, then A: “just as a sin offering sacrificed not for its own sake is unfit, so too a guilt offering sacrificed not for its own sake is unfit.”

But Rabbi Yehoshua rebuts this by bringing up a difference. With a sin offering, the blood of the slaughtered animal is sprinkled above the red line that encircles the altar at half its height: call this condition C. But a guilt offering does not meet this condition, since its blood is sprinkled below the red line. (The point of the red line seems to be precisely to introduce another variable in the sacrificial process, another way to distinguish more holy from less holy.) Perhaps, Yehoshua suggests, this difference corresponds to a difference in the rules governing intent, as well. He believes that if not C, then not A: since a sin offering and a guilt offering are not sprinkled the same way, maybe a sin offering and a guilt offering are also not the same with regard to the necessity for correct intention.

Rabbi Eliezer responds to the objection by pivoting to a different example. The Paschal offering, which is offered on Passover, is likewise sprinkled below the red line, and “if one slaughtered it not for its own sake it is unfit.” Here we find a case where we know that C does not hold, but A does. This seems to refute Rabbi Yehoshua’s suggestion that A is dependent on C: an offering can be sprinkled below the red line on the altar, and yet still be invalid if not sacrificed with the right intent.

But Rabbi Yehoshua has another objection to make. A Paschal offering also meets another condition, call it D: “its designated time is fixed.” That is, a Paschal offering can only be sacrificed at the appropriate time, on Passover, while a guilt offering can be sacrificed at any time. Thus while Paschal and guilt offerings are similar with regard to C (red line), they are different with regard to D (fixed time), and perhaps this difference means that they are also different with regard to A (intent matters). How do we know whether A depends on D?

Rabbi Eliezer responds by bringing up a case where A (intent matters) does not depend on D (fixed time): a sin offering can be sacrificed at any time, yet it is still disqualified by wrong intent. With a sin offering, D is false yet A is true. This suggests that with a guilt offering, too, A is true even if D is false. But this is only true if there is a valid analogy between sin and guilt offerings; and that is the very point that the argument had to prove in the first place. This is what English calls “begging the question,” and what Rabbi Yehoshua calls halila, going around in circles.

The failure of logical reasoning to resolve the issue forces Eliezer and Yehoshua to turn to the Torah, where Eliezer finds a seemingly ironclad statement in Leviticus 7:7: “As is the sin offering, so is the guilt offering.” Yet this is not the end of the debate. The rabbis go on to question other possible conditions for a sacrifice, asking whether they are shared by sin and guilt offerings. For example, a sin offering is brought for a transgression punishable by karet—roughly speaking, eternal punishment—while a guilt offering does not involve karet. Again, the blood of a sin offering “enters the innermost sanctum” of the Temple, since on Yom Kippur the High Priest offers a sin offering in the Holy of Holies; but the blood of a guilt offering does not enter the Holy of Holies. Should these differences mean that the sacrifices are different with respect to intent as well?

The reason why all this discussion is necessary, of course, is that the rabbis are trying to make sense of something that is fundamentally inscrutable: the will of God. Why did God order the Jews to offer these specific sacrifices in these specific ways? Which elements of a sacrifice are essential and which are inessential? The rabbis’ insistence on trying to answer such questions bespeaks their deep faith in human reason, and in the reasonableness of God. Everything he commands has a logic behind it, though it may sometimes prove too difficult for us to understand.


Adam Kirsch embarked on the Daf Yomi cycle of daily Talmud study in August 2012. To catch up on the complete archive, click here.

Adam Kirsch is a poet and literary critic, whose books include The People and the Books: 18 Classics of Jewish Literature.