Kinnim, Chapter 1, Mishnah 2. The first mishnah of the masechet established the basic rules of bird offerings: the bird chatat is performed on the lower portion of the mizbe'ach and the bird olah on the upper portion, and if either one is brought in the wrong place, the korban is invalid. Our mishnah now takes that principle and applies it to a practical problem: birds that became mixed up with one another, so that we no longer know which bird is which.
A chatat mixed with an olah
The mishnah opens: "chatat shenit'arva be'olah ve'olah bechatat" - a bird designated as a chatat became mixed together with a bird designated as an olah, or an olah became mixed together with a chatat. And the mishnah adds: "afilu echad beribo" - even one bird lost among ten thousand of the other kind. Even when one type is an overwhelming majority, the ruling is "yamutu kulam": all of them are left to die.
Why does the majority not simply swallow up the single bird? Because there is no bitul here. A living creature, a ba'alei chayim, is halachically significant, chashuv, and something of that stature is never nullified in a majority. Even one bird among ten thousand retains its own identity and imposes its doubt on the entire group.
The consequence is that every single bird in the mixture is now a safek. This one might be a chatat and it might be an olah, and the same is true of the next. Since a bird chatat brought above is invalid and a bird olah brought below is invalid, there is no place on the mizbe'ach where any of these birds can safely be offered. Not one of them can be brought, and birds of hekdesh that cannot be offered are left to die.
Background: a designated pair and an undesignated pair
Before we reach the next case, a background point that will accompany us through the remainder of the masechet. Someone who owes a pair of birds owes one chatat and one olah. If at the moment of consecration he stated which of the two would serve for the chatat and which for the olah, that assignment holds for good: the two may not trade roles, and each must be offered precisely as he declared it. The name for such a pair is ken meforeshet. And if the owner afterward loses track of which bird was which, we land squarely in the situation just learned, where neither of them may go up to the mizbe'ach.
Alternatively, a person may consecrate his two birds without saying anything at all about their roles. This is a ken setuma. Both birds are hekdesh, set aside for bird korbanot, yet at this stage neither has become a chatat and neither has become an olah. The roles are settled only afterward, by the hand of the kohen: whichever bird he offers as the chatat thereby is the chatat, and its companion turns into the olah on its own.
A chatat mixed into obligatory pairs
Now the mishnah presents the next case: "chatat shenit'arva bechova" - a bird that was definitely designated a chatat became mixed together with obligatory pairs that were left undesignated. The definite chatat is not nullified, and it may be brought only as a chatat. Meanwhile, in each undesignated pair, one bird is destined to serve as a chatat and one as an olah. So how many birds from this combined group can actually reach the mizbe'ach?
The mishnah answers with a tally: "ein kasher" - nothing here is fit for the mizbe'ach - "ela keminyan chatatot shebechova" - except as many birds as there are chatatot lodged within the obligatory pairs. That figure, and no more, may be brought.
Work it out with numbers. Suppose a bird that is unquestionably a chatat gets mixed into two obligatory pairs that were never assigned: five birds in the pile. The certain chatat is of course fit for chatat service, and within the pairs there are likewise two birds destined for chatat service, one drawn from each pair. Two chatat offerings may therefore be carried out from this group, and they will come out kosher whichever birds the kohen happens to pick up.
Follow the logic. If the first bird he takes is the definite chatat, it has been brought properly. If instead the bird he takes is one of the undesignated ones, that bird now becomes the chatat of its pair, and again the service is valid. The same holds for a second bird. But a third bird cannot be brought, because it may well be the partner of a bird already offered as a chatat, and that partner was destined to be the olah of its pair. Bringing it as a chatat, below, would ruin it. So with two pairs in the mixture the limit is two birds, which is precisely how the mishnah's rule protects us from an invalid korban. Many further permutations follow from this same reasoning.
An olah mixed into obligatory pairs
The mishnah then gives the mirror case: "vechen olah shenit'arva bechova" - and so too if a bird set apart as an olah became mixed into obligatory pairs that were left unassigned - "ein kasher" - nothing here may be offered - "ela keminyan olot shebechova" - except as many as the olot lodged in those pairs. The arrangement is exactly the one we just had, only that the certain bird which wandered in is an olah rather than a chatat, so the tally that limits us is the tally of olot.
The proportions make no difference
The proportions make no difference
The mishnah shuts the door on any appeal to relative quantity, spelling out three possibilities. "Bein shehachova merubah" - the unassigned obligatory pairs may form the bulk of the mixture, "vehanedava mimu'etet" - with the birds already earmarked as an olah the smaller part. "Bein shehanedava merubah" - or the picture may be reversed, the earmarked olah birds predominating, "vehachova mimu'etet" - and the obligatory pairs the few. "Bein sheshehen shavot" - or the two sides may come out even. However the numbers fall, the halachah does not budge.
And what is that rule in this last case? Since a definite olah is somewhere in the mixture, a chatat is entirely off the table. Any bird the kohen picks up might be that definite olah, and one may never bring a chatat out of doubt. What we do know with certainty is that we are holding olah material, either birds designated as an olah from the outset or the olah component of an undesignated pair. So olah offerings can go forward, but only up to the number that holds true in the worst case: half the birds of the undesignated pairs, and not one more. Go beyond that count and we have stepped into the land of safek, where the bird in our hands may really be a chatat, and a chatat brought above is invalid. The mishnah therefore keeps us strictly within the boundary of certainty, offering only those korbanot we know will be valid.