Bava Kamma Chapter 3 Mishnah 8. From here on we move on to the category of damage called 'Keren' - an animal acting in an unusual manner, with the intent to cause damage. There are two fundamental rules regarding it: the liability is to pay only half damages, and the obligation is limited to the value of the animal that caused the damage.
"Shenei shvarim tamim shechavlu zeh bazeh" - Two Tam oxen that injured each other:
Two neighbors whose animals - oxen, cows, or horses - began to fight and injured one another. In such a case, "meshalem bamotar chatzi nezek" - he pays half damages on the difference: we check who caused the greater damage and offset them. If one animal caused 100 in damages and the second caused 60, the difference is 40, and the owner of the animal that caused more damage pays half damages - 20.
"Sheneihem muadim" - Both of them are Muad:
Two neighbors whose oxen have a particularly bad temperament, and their owners were warned three times, until they were established as Muad. If they injured one another - "meshalmim bamotar nezek shalem" - they pay full damages on the difference: whoever caused more damage pays the difference in full damages. If one ox caused 100 in damages and the second caused 60, the difference is 40, and this amount is paid to the injured party.
"Echad Tam ve'echad Muad" - One is Tam and one is Muad:
Here we are dealing with a case where a Muad ox and a Tam cow injured one another. The owner of the ox, since it is a Muad, is liable for full damages, and the owner of the cow is liable for only half damages. We will first learn the Mishnah according to the Bartenura, who learns in accordance with the approach of Rashi:
"Muad betam" - the Muad is the one that caused more damage, and its owner is liable for full damages. In such a case, "meshalem bamotar nezek shalem" - he pays full damages on the difference: the Muad ox caused 100 in damages and the Tam cow caused 60, and in the final offset, the owner of the ox pays the owner of the cow 40 - full damages.
"Tam bemuad" - the Tam is the one that caused more damage. In such a case, "meshalem bamotar chatzi nezek" - he pays half damages on the difference: the Tam cow caused 100 in damages and the Muad ox caused 40. 100 minus 40 is 60, and since the cow is a Tam, its owner pays half damages - 30.
Seemingly, one could have calculated it differently: why wait and first offset the total damages, and not calculate the monetary liability of each party independently? In that same case where the Tam caused 100 in damages and the Muad caused 40 - the owner of the Tam is liable for half of 100, which is 50, and the owner of the Muad is liable for 40; 50 minus 40 is only 10, and not 30.
The practical Halachah:
The Bartenura and Rashi do not learn this way, but this is how Tosafos learns, and so is the opinion of the Rema and the ruling of the Shulchan Aruch. In Halachah, we do not look at the overall damage and the question of who caused more damage, but rather at the amount of the monetary liability, and we treat the two events separately. There is a great leniency in this: according to the approach of Rashi and the Bartenura, it could be difficult in certain cases to determine whether we are dealing with two separate acts or one combined act, whereas according to the Shulchan Aruch and the Rambam, it is always treated as two separate acts. Therefore: the owner of the Tam that caused 100 in damages is liable for 50, the owner of the Muad that caused 40 in damages is liable for 40, and the difference - only 10 - is what the owner of the Tam pays, unlike the Bartenura.
Two people who injured each other:
The same rule applies to two people who injure each other - for example, two who got into a boxing match and hit each other. If one is liable for 100 and the other is liable for 30, we offset the amounts, and he pays full damages for the difference: the one who caused damages of 100 will ultimately pay 70, which is 100 minus 30.
"Adam bemu'ad, mu'ad be'adam" - a person who injures a mu'ad, or a mu'ad that injures a person:
This refers to a person and a mu'ad animal - such as a lion that a person attacked and injured, and the lion returned the attack and injured him. The owner is liable for the damages caused by the lion, and the person is liable for the damage he caused to the lion, and he pays full damages for the difference: the one who caused more damage pays full damages based on the difference.
"Adam betam, tam be'adam" - a person who injures a tam, or a tam that injures a person:
Here we are dealing with a person and a tam cow that injured each other, where a person is always considered mu'ad:
"Adam betam" - the person caused more damage, and therefore "meshalem bemosar nezek shalem" - pays full damages for the difference: the person caused 100 in damages to the cow, and the cow caused 80 in damages to him; 100 minus 80 is 20, and the person pays 20.
"Tam be'adam" - the cow caused 100 in damages to the person, and the person only caused 80 in damages to it; 100 minus 80 is 20, and the owner of the tam cow pays half damages - 10, according to the Bartenura.
But according to the approach of the Rema and the Shulchan Aruch, the calculation is completely different: the tam caused 100 in damages, and therefore its owner is liable for 50; the person caused 80 in damages to the tam, and therefore he is liable for 80. 50 minus 80 - it turns out that the person is the one who is liable, and he must pay 30.
The approach of Rabbi Akiva:
Rabbi Akiva says "af tam shechaval be'adam - meshalem bemosar nezek shalem" - even a tam that injures a person pays full damages for the difference. In his opinion, the law of half damages for Keren was only stated when the animal damages a person's property, but when it injures a person's body, the payment is full damages, and this he derives from another source. Therefore, a tam cow that attacked a person and caused 100 in damages - its owner is liable to pay 100 and not 50.
The halachah:
The halachah follows the Sages, that for the damages of a tam, half damages are paid in any case, whether to a person or his property. However, it should be noted that even according to Rabbi Akiva, although full damages are paid, the payment is limited to the value of the animal that caused the damage - meaning that even according to his approach, the collection is only from its body.
In summary: In this Mishnah, we learned the laws of mutual damages for Keren: two tam animals and two mu'ad animals that pay for the difference, one tam and one mu'ad in both directions, two people who injure each other, and a person and an animal - tam or mu'ad. We discussed the two methods of calculation: the approach of Rashi and the Bartenura, which first offsets the damages, and the approach of Tosafos, the Rambam, the Rema, and the Shulchan Aruch - that each incident is judged independently and the difference is between their monetary liabilities, and this is the halachah. Finally, we discussed the approach of Rabbi Akiva regarding a tam that injures a person, and that even according to his approach, the payment is collected from the body of the animal alone.