Wednesday, May 09, 2007

Daf Yomi Yevamot 13a: Rav Ashi's Proof By Induction

As I was preparing Rif Yomi, I noticed a proof by induction. In induction, you prove a base case, and then show that if something is true for n, it is true for n+1 as well (the inductive step).

In terms of rivals and rival's rivals, there are two approaches. A general derasha to cover all, or an induction. To cite the gemara:
{Yevamot 2b}
כיצד פוטרות צרותיהן היתה בתו או אחת מכל העריות האלו נשואה לאחיו ולו אשה אחרת ומת כשם שבתו פטורה כך צרתה פטורה
הלכה צרת בתו ונישאת לאחיו השני ולו אשה אחרת ומת כשם שצרת בתו פטורה כך צרת צרתה פטורה ואפי' הן מאה

How do they exempt their rivals? If his daughter, or any one of these forbidden relatives, was married to his brother, and he {=his brother} had another wife, and he {=his brother} died, just as his daughter is exempt {from requiring yibbum or chalitza from him}, so too her rival is exempt.
If the rival of his daughter went and married {via yibbum} his second brother, and he {=his second brother} had another wife, and then he {=his second brother} died, then just as the rival of his daughter is exempt, so it the rival of his rival exempt {from yibbum or chalitza with him}, and even if they {the levels} are 100.

{Yevamot 13a resumes}
מה"מ אמר רב יהודה אמר קרא לצרור התורה ריבתה צרות הרבה אצל ערוה אחת
רב אשי אמר צרת צרתה לא צריכא קרא סברא הוא צרה מאי טעמא אסירא דבמקום ערוה קיימא צרת צרתה נמי במקום צרה קיימא:

From where do we know these words?
Rav Yehuda said: The verse stated {Vayikra 18}: לצרור. The Torah thus includes many rivals by one forbidden relation.
Rav Ashi said: The rival of the rival does not require a verse. It is logic. A rival, what is the reason she is forbidden? Because she stands {together} in place of the forbidden relation. The rival of a rival also, she stands {together for yibbum} in place with the rival.
Thus, there is a general rule that if someone is a rival with someone forbidden, they are forbidden {and thus exempt from yibbum and chalitza as well}.


littlefoxling said...

I don't agree with you this is a proof by induction. But, before I write a long post explaining why I disagree with you, let me first clarify, is this post just a joke or do you really mean this is induction

joshwaxman said...

i meant it seriously, and think there is a recursive step.
I look forward to your post.
Kol Tuv,

littlefoxling said...

It would seem to me that the chidush of Rav Ashi is not that you can use induction here, but rather that tzras tzaraso is also considered bimakom ervah. This assumption is not obvious. It’s true the tzarah is “eishes uchiv shelo bimakom yibum” which would ordinarily be considered a “chiyuv kures.” But, is it so obvious that the tzarah, who is only putur from yibum has a din of an erveh in the sense that she can be poteres trarasah? In other words, I’d argue that the debate between Rav Ashi and the Rav Yehudah is not whether induction is a valid mathematical proof, but rather, whether or not the sevorah of tzras erveh is really relevant to tzras tzraso.

I’d muster the following support to my position

1. According to you, what is Rav Yehuda’s sevorah? How can one deny induction?
2. The discussion in the gemorah seems to revolve around the second tzurah. “Tzras Tzarasah” itself, the second case itself, where no fancy shmancy induction is needed is the case Rav Ashi is discussing.
3. This case of induction is quite obvious, it’s hard to believe that this is Rav Ashi’s chidush.

In fairness, I should note,
1. I see what you are saying and am not really sure of what I am saying
2. Even according to me, Rav Ashi’s proof is technically induction, I’m just saying that’s not his chidush, he’s not coming to tell you that you can use induction, nor does Rav Yehuda deny there is such a thing. Induction in this situation is obvious, and it is a given. Rav Ashi’s chidush is that saying n implies n +1 is true here. But, the fact that once you’ve established n implies n +1 you can use this as induction is not his point. In other words, I guess I’m saying he’s not thinking in those terms.

If I remember from my frum days, the chakirah I posed above has lots of implications for many gemorahs in yevomos, but I don’t really remember much specifics from that bygone era. To quote Rav Elazar:

“Hacheresh Hayu Libum”

joshwaxman said...

interesting points. I'll have to think about it.

My first reaction would be that
I agree with you as per your point #2 in "in fairness."

off the cuff, I would say:

1) I don't think that Rav Yehuda would deny induction or Rav Ashi's proof. Rav Ashi was a later generation. Who says that he would not say "ein hachi nami," that is another good way at obtaining these results? Nor am I so convinced that Rav Ashi necessarily discards Rav Yehuda's derasha (such that he needs it for something else).

2) The Mishna at least (which is cited by the geonic pesikta כיצד פוטרות צרותיהן וכו), after mentioning tzrat tzarata, says "until 100 levels," so they are equivalent, and so it is just shorthand.
Indeed, these are the last words cited by the Rif in the Mishna, who assigns it there. (though of course assigning Mishnayot interspersed with gemaras is post-Talmudic.) I'm not sure what you mean by the gemara revolves around the second tzara.

3) I don't know what is or what is not obvious. Both elements (that a rival stands as a primary ervah and that *as a result* rival's rival = the original rival in this respect) are possibly novel components.

This is not to say that others would disagree with the induction.


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