Yosef, this is a brilliant question!

I think it is the question of the sefer Birkas ha'Zevach, that there are only two Kal v'Chomers. I will explain, bs'd.

If we look on the side of the Gemara 50b, where the Gemara asks, "Can a Kal v'Chomer teach through a Kal v'Chomer," there is a note that says עי' רש"ק -- "See Rashak." "Rashak" stands for Rebbi [Aharon] Shmuel Kaidanover (1614-1676), the author of Sefer Birkas ha'Zevach.

The question of the Birkas ha'Zevach goes like this. We are trying to learn C from B with a Kal v'Chomer. Whenever one learns a Kal v'Chomer, the matter that one is learning from possesses some leniency and even so a strict Halachah applies to it. In other words, B has a Kal aspect, and if a certain Chumra applies to B, this Chumra will certainly apply to C which is generally more stringent than B.

The Birkas ha'Zevach raises a difficulty. Since B has a Kal aspect, how could we have learned the Kal v'Chomer from A to B in the first place? One cannot learn a Kal v'Chomer from A to B if B is more Kal than A! So what must we say? We must say that the Kal aspect that B possesses is also possessed by A. Since B is not more Kal than A, this means we can learn a Kal v'Chomer from A to B.

But now a new question arises. Since A and B both have a Kal aspect that is not possessed by C, why does one have to pay attention to B at all? One could skip B and learn a Kal v'Chomer directly from A to C!

I think this is your question -- why do we need B>C, when we can use directly A>C?

The Birkas ha'Zevach does not answer his question, so we will need to think about it.

Yeyasher Kochacha!

Dovid Bloom

I was looking again at your question, Yosef, and I do think that what is bothering you is the question of the Birkas ha'Zevach, because you wrote at the end:

"A>B B>C therefore A>C"

which is the same as the question of the Birkas ha'Zevach; namely, that why do we need B at all, since we can say straight away A>C?!

An answer to the question of the Birkas ha'Zevach was given by his student, the Panim Me'iros, who gives an example where the first Kal va'Chomer cannot automatically be carried to the third without the help of the second. In other words, A requires B in order to reach C, because A cannot always go directly to C.

The example is from the Mishnah below, end of 69b. Somebody performed Melikah on a bird and it was found to be a Tereifah. Melikah is a way of slaughtering a bird (with the fingernail which cuts the bird's neck) which is only acceptable in the Beis ha'mikdash. The question is, does this Melikah on the Tereifah remove the impurity from the bird?

The Mishnah (69b) gives a Kal va'Chomer: The carcass of an animal renders Tamei anyone who touches or carries it, but if it was slaughtered this removes the Tum'ah even if the animal was found to be a Tereifah. The carcass of a bird possesses less severe Tum'ah than an animal, since it does not make Tamei a person who touches or carries it, but only somebody who eats it.

Now we say the Kal va'Chomer: If the Shechitah of the animal removes its severe Tum'ah (even if it transpired that it is a Tereifah), then the Shechitah on the bird surely can remove its lighter Tum'ah (even if it transpired that it is a Tereifah).

We have therefore established that the Shechitah on a bird removes its Tum'ah even if it is a Tereifah.

The next question is: Can the Melikah on a bird remove its Tum'ah? Rebbi Yosi (in Mishnah 69b) says that we have gone far enough and cannot go any further. Rebbi Yosi holds that Melikah cannot remove the Tum'ah.

The Panim Meiros explains that the reason that this cannot happen is because we are starting off with the Shechitah of an animal and trying to conclude with the Melikah of a bird, but the problem is that there are too many differences between the starting point and the finishing point. We started with the Shechitah of an animal, made one change to get to the Shechitah of a bird, and then made a second change in order to get to the Melikah of a bird.

This is stated by Tosfos to Zevachim (50b-51a):

- The reason we cannot learn from Shechitah of an animal to Melikah of a bird is because thee are two differences: Melikah is different from Shechitah and a bird is different from an animal.

Therefore, one has to go step by step. One cannot simply say A>C but have to say A>B then B>C and only then A>C.

Kol Tuv,

Dovid Bloom