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What is the process involved for figuring out the halacha of today based on something written in the Talmud?

Basically, what does a modern day posek do starting from the Talmud to reach a halacha he can put in a sefer, such as mishna berurah.

Aside from the arguments within the Talmud, there are also arguments between rishonim and acharonim. That being the case, it's not so clear how to actually arrive at any given halacha lemaaseh.

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While agreeing with all that is mentioned here there is nothing like putting it into practice. One should choose a small self-contained gemoro it doesnt matter where, and try and learn it with all rishonim. The mesivta gemoro is useful for this. Then if on orach chaim follow the mishne berura. Hebrewbooks.org has all the seforim he had (and more). Use them all and then go to the mishne berura. The new MB also includes everyone else as well like the Aruch Hashulchan. If done properly, that would be called real learning. Otherwise it is just 'reading'. –  preferred Jun 23 at 14:38

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From the Gemara, one would learn the relevant Rishonim, Achronim, Teshuvah literature, and later Halakhic works... there's quite a lot to go through on any given 'sugyah' (Talmudic topic).

Regarding the choice that a posek is faced with when there are differing opinions on the matter, there are a few general rules aptly referred to as 'kelalei hapsak' (general rules of paskening) which are scattered throughout the Gemara and Rishonim (like any halakha) and have been collected by certain Achronim, such as by the Peri Megadim, Knesses Hagedolah, and the Maharit Algazi, and more recently, Rav Ovadia and Rav Yitzchak Yosef (in the introductions to Yalkut Yosef, and in a major three volume work called 'Ein Yitzchak').

These rules are varied and detailed, but some major ones are: following the majority (not necessary a quantitative majority; some 'greater' earlier poskim might outweigh numerous others), following the Shulchan Aruch and Rama, and following well established customs. Different poskim will give different weights to these considerations, for example, Rav Ovadia Yosef gives much stronger weight to the Shulchan Aruch. Of course, there's also the question of what makes the most sense. Rav Moshe Feinstein was wont to argue on earlier poskim becuase he thought he had a better explanation of a Gemara than they did, for example. These 'shikulim' (measuring/weighing) aren't clear cut and different people feel differently about these rules, which is why we follow poskim (as opposed to making our own decisions) in the first place; only someone with the appropriate breadth and depth of knowledge can be assumed to have the proper instincts about when and how to apply these rules.

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I'm going to disregard your second paragraph, since a "modern-day" posek (your example was of the Mishna Berurah) doesn't go from the Talmud directly into halakha lema'aseh, but is already familiar with a great wealth of material that both predates the former and spans the two. For regular people, like ourselves, who want to gain a deeper understanding of the relationship between Talmudic dicta and contemporary halakhot, there are ways in which one can do this.

Let's assume that you are commencing with an individual sugya. The chances are that any Shas that you are using is going to have a useful index called Eyn Mishpat, which was compiled by a 16th century scholar named R' Yehoshua Boaz Barukh (known as the "Shiltei haGibborim"). That provides references to passages in the Rambam's Mishne Torah, R' Moshe of Coucy's Sefer Mitzvot Gadol ("SMaG") and R' Yaakov ben Asher's Arba'ah Turim. The latter work employs the same numbering system as R' Yosef Caro's Shulchan Arukh, giving you references to that corpus as well.

How you proceed from there is up to you. Personally, I prefer to use the Arba'ah Turim, since the commentaries of R' Yosef Caro ("Bet Yosef") and R' Moshe Isserles ("Darkhei Moshe") will give you a wealth of information that you can then follow up separately. Furthermore, since it employs the same numbering system as the Shulchan Arukh, you can proceed next to the Arukh haShulchan (by R' Yehiel Michel Epstein, a 19th-20th century scholar) and find references there to an abundance of material published after the 16th century. Similarly, if the material that you are looking at is in Orach Chayim, you can check the Mishna Berurah for the author's Be'ur Halakha, which is similar in scope.

If your interest leans more towards the Rishonim and the earlier period of Acharonim, I would advise checking the Mordekhai and (if what you are looking at is one of the masekhtas in which it appears), the Shittah Mequbetzet. The former was by the students of R' Mordechai ben Hillel (13th c.), and the latter by R' Bezalel Ashkenazi (16th c.). Where the former is a source for a large number of statements by Rishonim not recorded elsewhere, so too is the latter for early generations of Acharonim. Even more easily, you can follow Eyn Mishpat to the Rambam and check the commentaries there (Kesef Mishna, Hagahot Maimoniyot, etc).

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