Generally speaking, what is the Hasidic approach to making halachic rulings, and how does it in general differ from the Yeshivish and Modern Orthodox methodologies. I understand the Alter Rebbe of Chabad Lubavitch Shneur Zalman ZT”L of Liadi wrote the Shulchan Aruch Harav even though Ashkenazi commentaries on the Shulchan Aruch existed, in order to reflect Hasidic responsa, how did his methodology differ from the non Hasidic approach (both Yeshivish and Modern Orthodox).
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The "Rav's shulchan aruch" predates chasidic responsa literature. You ask a fair question, and there well may be differences in aproach, but it must be remembered that the basic approach is overwhelmingly similar. While the Rav's Shulchan Aruch may indeed be more influencial in Hasidic circles, it is quoted often (sometimes?) by Mishna Brura. MB is generally #1 authority for halacha among "Yeshivish", and among many Chasidim too.– Naftali TzviCommented Oct 17, 2023 at 0:29
1 Answer
As far as the Shulchan Aruch HaRav, the Chabad website here provides a good guide as to Rav Shneur Zalman of Liadi's approach and his mindset when authoring it:
What goal did Rabbi Schneur Zalman set for himself in this work? In truth, it was not he who set the goal. Rabbi Schneur Zalman, executed the project that his own teacher, Rabbi DovBer “the Maggid” of Mezeritch, had charted for him. The following passage—which appears in the introduction to the Shulchan Aruch ha-Rav, signed by Rabbi Schneur Zalman’s three sons —is illuminating:
Since the needs of the Jewish people are many, especially in these difficult times, with rising expenses and the encumbrance of earning a livelihood burdening each individual—with his soul he acquires bread—and they have little mind to engage in lengthy study of the sea of Talmudic and legislative sources, to know the source of the law and specifically rulings with their justification. Even great scholars who have ability and fame for their knowledge of the Talmud have difficulty arbitrating between the various legislative authorities when studying to derive a final ruling in agreement . . . for in most areas there are divided opinions . . .
Therefore, from heaven it was decided, via the aforementioned holy rabbi [Rabbi DovBer, the Maggid of Mezeritch], to search assiduously amongst his disciples to find an individual with the spirit of G‑d within him and the capacity to understand and to ordain clear legislation . . . to organize all the established laws that are recorded in the Shulchan Aruch and all the later authorities in concise language, with reasoned justification. He chose his honor, our holy master and father [Rabbi Schneur Zalman], and urged and persuaded him, saying, “There is no one with understanding and wisdom like you, able to descend to the depth of the halachah and do this work, a work of holiness, to bring to light the concise essence of the reasons for the rulings mentioned, including all the words of the earlier and later authorities, refined sevenfold, each matter in its correct place without confusion and muddle, arriving at the conclusive ruling that becomes clear and emerges from the words of all the legislators until the scholars of today . . .
In short, there are three fundamentals that Rabbi Schneur Zalman had in mind as he set to work on his Shulchan Aruch: organization (“without confusion and muddle”), explanation (“the reasons for the rulings mentioned”) and arbitration (“the conclusive ruling that emerges from the words of all the legislators”).
See more there for further detail.