1) THE NECESSITY FOR THE SIGNATURES OF WITNESSES ON A GET TO BE "LISHMAH"
QUESTION: The Gemara concludes that the reason why the Tana of the Mishnah requires that both the Kesivah (writing) and the Chasimah (signing) of the Get be done Lishmah, according to Rabah, is that he follows the view of Rebbi Elazar. Although Rebbi Elazar says that Edei Chasimah (witnesses signed on the Get) are not required altogether, nevertheless if witnesses do sign he Get, those witnesses must sign Lishmah, and if they do not sign Lishmah the Get is invalid. (Rebbi Elazar issues a similar ruling in the case of a invalid witness (such as a relative) who signed a Get. Although he maintains that witnesses do not have to sign a Get, invalid witnesses signed on a Get are worse than no witnesses signed on a Get, and the Get is invalid.)
What is the logic behind this ruling? When witnesses sign the Get she'Lo Lishmah, why is the Get invalid? It is understandable that when an invalid witness signs the Get, there is justification for the Chachamim to invalidate the Get lest people mistakenly infer from the fact that this person signed the Get that he is a valid witness (and use him as a witness for other matters). However, when a Get is signed she'Lo Lishmah, what possible mishap could result for which the Chachamim should invalidate the Get? (The witness himself is a valid witness for all other matters, and he signed in attestation of what actually occurred; he merely did not sign Lishmah.) Certainly, mid'Oraisa there is no reason for the Get to be invalid since the Torah does not require Edei Chasimah in the first place.
ANSWERS:
(a) TOSFOS (DH Modeh) explains that the Rabanan invalidated a Get that is not signed Lishmah because people might mistakenly assume that the Kesivah also does not need to be Lishmah. Although this Get was written already, the next time they write a Get they will assume that they do not need to write it Lishmah. (They will not realize that the Edei Chasimah are entirely extraneous to the validity of the Get.)
(b) The RASHBA suggests that there is a case in which the Get is invalid mid'Oraisa because the Edei Chasimah did not sign Lishmah. In a case in which the Get was handed over in front of Edei Mesirah (witnesses who see the giving of the Get) but those witnesses are no longer available (they died or left the country), the onus of validation falls on the Edei Chasimah, the witnesses signed on the Get. Rebbi Elazar says (3b) that the Get may be validated through the Edei Chasimah in such a case. How do the Edei Chasimah prove that the Get was delivered in front of witnesses (Edei Mesirah)? Since their signatures testify that the Get was not forged, it is assumed that everything about the Get was done in the proper manner.
The Rashba writes that when the Get is validated through the Edei Chasimah, their signatures and testimony become an integral part of the Get since Beis Din authenticates the Get based on their signatures. Therefore, just as every other important part of the Get must be written Lishmah, the signatures -- through which the Get is authenticated -- must also be written Lishmah. In such a case the signatures are considered part of the Kesivah.
This is why Rebbi Elazar says that when witnesses sign the Get she'Lo Lishmah, the Get is invalid mid'Rabanan. The Chachamim were concerned that at a later time Beis Din would rely on the Edei Chasimah to validate the Get, and if the witnesses did not sign Lishmah, the Get cannot be validated on the basis of their signatures.
(c) The Rashba points out that according to the RIF, signatures that are no Lishmah might invalidate the Get mid'Oraisa. The Rif maintains that according to Rebbi Elazar, although the Edei Mesirah empower the Get to be effective ("Edei Mesirah Karti"), Edei Chasimah also give the Get validity without Edei Mesirah. (See TOSFOS, DH d'Kaima Lan.)
The Rashba suggests that according to the Rif, Rebbi Elazar requires that both the Edei Mesirah and Edei Chasimah be Lishmah, since they are of equal importance. (The Rashba concludes that he does not favor this approach. If Edei Chasimah are included in the words "v'Kasav Lah," then every Get should require Edei Chasimah, and yet Rebbi Elazar permits a Get to be written without Edei Chasimah.)