The kuntres argues that the clearest halachic statement of the Gra on the number of Techeiles strings is his Biur HaGra to Shulchan Aruch. There he explains the sugya at length and concludes that Techeiles and white are equal: out of the four original strings, two are Techeiles and two are white, producing four blue and four white ends after they are folded. Barkin therefore argues that this should not be displaced simply because elsewhere we find a different-looking hagahah or textual comment.
R’ Eliyahu Tavger’s main argument is that the Gra later changed his mind. In the Gra’s hagahos to the Sifrei, he corrected the text to a reading that says, in effect, three strings of white and a fourth of Techeiles, which fits the Raavad’s ratio of one full Techeiles string out of four. Tavger therefore understands this as evidence that the Gra’s final shittah was the Raavad.
Barkin’s answer is that this assumes too much about what a hagahah to the Sifrei means. The Sifrei is itself an ancient Chazal text from roughly the Tannaitic/Gemara period; the Gra was obviously not “correcting the Sifrei” in the sense of disagreeing with Chazal. He was trying to reconstruct the original wording of a text that had passed through many centuries of copying and printing. There were already competing girsaos of this passage. One version, reflected by Tosafos, had four Techeiles and four white; another transmitted form could be understood as three white and a fourth Techeiles. The Gra apparently judged that the latter was the correct original girsa and emended the printed Sifrei accordingly.
That is an important distinction. Determining what the Sifrei originally said is not automatically the same thing as paskening like that Sifrei against the Bavli or against one’s own halachic analysis. The appendix of the kuntres stresses precisely this point: the Gra’s hagahos were often extremely abbreviated and could have been written to establish the correct girsa, explain the position of a particular Rishon, or make a textual observation—not necessarily to issue a final psak.
In fact, the issue of textual corruption is very real here. The different versions contain small changes such as “ארבעה” — four — versus “רביעית” — a fourth, exactly the sort of change that can radically alter the meaning while arising relatively easily in textual transmission. Barkin even notes that the wording “three … and four” can itself be a legitimate Chazal-style expression meaning “three … and a fourth,” so the manuscript evidence is not as simple as saying one version obviously represents a different halachic tradition.
Barkin therefore says that the later Sifrei correction does not prove that the Gra retracted his psak in the Biur to Shulchan Aruch. If the Gra had actually changed his halachic position on something with practical consequences, Barkin argues, one would expect stronger evidence than a terse textual hagahah—especially since the Gra never erased or corrected his detailed discussion in the Shulchan Aruch. He brings the Chazon Ish to support the general point that where the Gra actually retracted something relevant to halachah, one would expect the earlier statement to be removed or corrected.
So the real machlokes in the kuntres is not whether the Gra wrote the Sifrei hagahah—he did. The question is what that hagahah proves.
Tavger reads it as: the Gra discovered the correct Sifrei and therefore changed his mind to the Raavad.
Barkin reads it as: the Gra was correcting the ancient text of the Sifrei; that tells us what girsa he considered original, but it does not by itself overturn the explicit halachic reasoning he left intact in his Biur to Shulchan Aruch.
And this fits the larger practical point made by R’ Yisroel Belsky in his haskamah: quite apart from resolving every detail of the Gra, the overwhelming majority of the Rishonim discussed by Barkin follow the two-and-two arrangement of Rashi and Tosafos, and the Mishnah Berurah paskens that way as well.
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