The Mishnah (4:6) begins by teaching that one is not obligated to support their children. The Mishnah continues that this was derived from the condition placed in the ketubah that if the father passes away, the daughters are supported by the estate until they marry, while the sons share the mother's ketubah (aside from the rest of the estate). Since the clause regarding the sons' inheritance is only applicable after the father's death, the same is true regarding the daughters' support – it only applies after the father's passing.
The Bartenura explains that the ruling in the Mishnah both depends on the age of the children and the financial situation of the father. If the children are younger than six, then Beit Din force the father to support his children and even forcibly extract the funds to do so. The Mishnah is referring to where the children are ages six and above. In that case, if the father is a financially capable of supporting his children with ease, then Beit Din would similarly force him; much like Beit Din have the capacity to force other members of the city in matters of tzedakah. If the father however is not wealthy, while the Beit Din would take care of the children's needs, they would also publicise the cruelty of the father in not supporting his children.
We find that the supporting one children who are six and above stems from the mitzvah of tzedakah. That being the case, the Tosfot ask how Beit Din can force the wealthy father? The Gemara (Chulin 110b) explains that Beit Din do not force one to perform a positive mitzvah whose reward is presented along with its command. The Tosfot provide several answers. First, they suggest that in this case the funds are not be forcibly extracted. Instead the "force" being used is verbal.
The Ran however finds this answer difficult, given that if the father is not wealthy, verbal pressure is also used. If the forcing in the case of the wealthy father is also verbal, then there appears to be no real difference whether the father is wealthy.
The Tosfot also suggest that the forcing is after the citizens have determined the amount of money that should be collected to support the children. In that case they can force the father to contribute based on their evaluation.
Finally, the Tosfot suggest that in this case Beit Din can force the father to pay, given that there are negative prohibitions that are violated if they avoid giving tzedakah.
Returning to the Tosfot's question, the Ketzot, citing the Kesef Mishnah, argues that since the mitzvah of tzedakah is a Torah command, there exists a shi'abud – a financial lien on one's property. Consequently, when Beit Din forcibly extract the funds, it is equivalent to Beit Din collecting a debt he owes to the poor – we are simply collecting what is rightfully theirs. In short, this case is not to be confused with forcing one to perform a positive mitzvah as this mitzvah has an existing financial lien.
Why then do the Tosfot not present this answer? The Kehilot Yaakov suggests that the mitzvah of tzedakah is different. It is true that for a positive mitzvah there is exists a shi'abud such the monetary obligation can be forcibly recovered. The Kehilot Yaakov cites Arachin and Pidyon Ha'Ben as examples. In this case however, the mitzvah of tzedakah is different in that is it is defined as a case of mamon she'ein alav tovin – a monetary case that does not have a defined claimant. This is because one can choose to whom they wish to give tzedakah. He cites the case of one who damages one of the matanot kehuna (gifts to the kohanim) that have not yet been given to a kohen. The Gemara (Bava Kamai 39a) rules that the one is exempt since it a case of mamon she'ein alav tovin. The mitzvah of tzedakah would be similarly defined as a case of mamon she'ein alav tovin. This then explains why the Tosfot did not present the Ketzot's.1
1 The Kehilot Yaakov notes that pikyon ha'ben would also appear to be a case of mamon she'ein alav tovin as one could perform the mitzvah with any kohen. He cites the Ritva who explains that that case here is referring to "makirei kahuna" – where the family has an existing arrangement to give their matanot kahuna to a specific kohen. Consequently, that case would no longer be defined as mamon she'ein alav tovin. The Kehilot Yaakov also provides other explanations why the case of pidyon ha'ben is different. See inside.
Note that the Kehilot Yaakov continues explaining that with tzedakah a shi'abud does indeed exists. This is in the context of the ani'im in one's city where the gabbai acts as the shaliach – there is a claimant. It is regarding this obligation to give each ani a small amount that the Kehillot Yaakov suggest the Kesef Mishnah is referring (and not our case).
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