Bava Kama 47a: If one man brought his ox into another man’s front yard with permission, and the house owner’s ox gored it, or his dog bit it – the house owner is liable. Rebbi says: He is not liable unless he took upon himself to guard it.
Shulchan Aruch Choshen Mishpat 398:5: The Mechaber rules like Rebbi and the Rema rules like the Sages (the anonymous first opinion in the Mishnah).
בבא קמא מז ע”א: הכניס שורו לחצר בעל הבית שלא ברשות, ונגחו שורו של בעל הבית או שנשכו כלבו של בעל הבית ־ פטור וכו’ ואם הכניס ברשות ־ בעל החצר חייב, רבי אומר: בכולן אינו חייב עד שיקבל עליו לשמור.
חו”מ שצ”ח ס”ה: ואם הזיק שור בעל החצר לשור הנכנס ברשות פטור, אלא אם כן קבל עליו בעל החצר שמירתו. הגה: ויש אומרים דכיון דנתן לו רשות ליכנס הוי כאלו קיבל עליו שמירתו. (טור בשם ר״י והרא״ש) .
A woman in Woodridge, New York had a ferocious pitbull, and her grown son also had a dog. The two dogs did not get along, so whenever he visited her, she was careful to keep the pitbull locked up. But one time she forgot and left it loose. It immediately attacked the other dog, injuring it badly. She asked Rabbi Hillel Grossman, the rav of Woodridge, if she had to pay.
The question can be broken down into two issues: 1) Is the halacha like the Mechaber or the Rema? 2) Even if the halacha is like the Mechaber, maybe since both of them knew that the pitbull would be likely to attack the other dog, the understanding between them when the son entered was that the mother would lock up her pitbull, so it is considered that she took upon herself at least this level of caution.
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