Kinnim, Chapter 3, Mishnah 3. Our chapter continues to follow the kohen who did not turn to a chacham before acting. Earlier in the masechet we learned the rule for a bird chatat that became mixed up with a bird olah when we can no longer tell which is which: both are left to die, since we have no way of knowing where on the mizbeach each one belongs. But that ruling describes the case where the question was brought to a chacham beforehand. Our mishnah asks what the law is after the fact, when the kohen never asked anyone, and simply went ahead and offered the birds as he saw fit.
The Case
The mishnah presents the mixture: "chatat lazo ve'olah lazo", this woman is obligated in a bird chatat and that woman is obligated in a bird olah. The birds became confused with one another, and we cannot identify which bird carries which designation. Whatever other birds these women owed have already been brought properly. What remains is one chatat and one olah, and no way to tell them apart.
Recall the basic distinction that governs everything here. A bird olah is offered above the red line that runs around the mizbeach, and a bird chatat is offered below it. Offer either one in the wrong place and it is invalid. Our kohen, acting on his own, has to choose.
Three Ways He May Have Acted
The first possibility: "Asah chulan lema'alan", the kohen performed the service of both birds in the upper zone, treating each of them as though it were an olah. Here the ruling splits: "mechetza kasher", one portion of his work stands, and "mechetza pasul", the other portion falls away. Whichever of the two birds genuinely carried the status of olah received its blood service precisely where an olah requires it, so that woman's obligation is discharged. The second bird, the one that was truly a chatat, had its service done in the upper area, a location that disqualifies a bird chatat, and it is therefore unfit.
The second possibility runs the other way: "Chulan lematah", he performed both services beneath the line, in the zone reserved for the bird chatat. The outcome mirrors the first case, half fit and half unfit. The bird that was in reality the chatat was handled in its proper zone and counts, while the bird that was in reality the olah was brought in the lower area instead of the upper one, and an olah handled there is disqualified.
"Chetzyan lema'alan vechetzyan lematah." Here he split them, offering one bird above the red line and one bird below. At first glance this looks like the ideal outcome. One bird was treated as an olah and one as a chatat, which is precisely what the two women owe. Nevertheless the mishnah rules "sheteihen pasul", both are invalid.
Why Both Are Invalid
The mishnah supplies its own reasoning with the words "she'ani omer", meaning that I am within my rights to propose the following scenario: the bird whose service was performed in the upper zone may have been the chatat, "chatat kerva lema'alan", while the bird handled in the lower zone may have been the olah, "ve'olah lematah". Should that be what actually occurred, the chatat was serviced in a spot that ruins it, and the olah was serviced in a spot that ruins it too, leaving nothing of value in either offering.
In other words, since we cannot identify the birds, we must reckon with the worst possible reading of what the kohen did. Perhaps each bird was offered in exactly the wrong place. Because that possibility cannot be ruled out, neither woman has fulfilled her obligation, and both offerings are invalid.