As explains the Ramchal in the end of the third chapter of Derech Tevunot, one needs to translate the Gemara language in a formal language, formal from some point of view. To make each sentence clear.
In the second chapter he explains that we need to find the concatenation between of statements. There are 7 types of statements from the dialectical point of view. To be short, affirmation, question, answer, objection, solution, proof, counter proof.
Gemara has a semi formal language, there are key words, as parich, פריך, metive, מתיבי, urminehi, ורמינהי, Tayna, תניא, matkif, מתקיף,you can learn them easily in Sefer Halichot Olam.
There are also rules of Hermeneutic, also well explained in H.O. But it's not formal as it seems.
Comparison between cases needs also a particular effort.
The Darkey Hatalmud from Rabbi Yitschak Kampenton explains that a great effort to understand the thought that underlies each opinion,objection, and the change between the thought of objections and solutions, the evolution of the opinions. He enhances the importance of thinking about why Rashi rejected a Girsa, why he doesn't explain as Tosfot. In Tosfot there are also key words.
The decision making of the Gemara is rarely based on binary formal logic. Because generally each opinion is logically coherent. Mostly, the decision is to follow the most convincing opinion or the most sustained by previous halachic authorities.
The talmudist need to know each opinion in the topic and to memorize it, e.g. Abbaye opinion despit that it's not the Halacha need to be memorized because in the next topic you need to check how Abaye is coherent with himself. If you find that he isn't, this show that something is lacking in your understanding. So, the sugia need always be learned in multiple points of view, as a local suggia and as coherent with other suggiot in the Talmud, linked to this. This fact is particularly obvious in dine Mamonot. This kind of learning is acquired by learning Tosfot, or at least Tosfot Yom Tov in Mishnayot. After you acquired a good level you can look the Rabbi Akiva Eiger, who emphasizes treatment of contradiction between suggiot.
When the Gemara is decifered, the logical structure is often very long, but not hard to grasp at all. There is no need for syllogistic knowledge.
We can note that Ramchal in Derech Tevunot and Sefer Hahigayon doesn't address seriously study of syllogisms. Gemara itself doesn't formulate syllogisms because the logic of Gemara is not enmeshed.
The work of talmudists is mainly to decifer the form and to evaluate the content. For this, formal algorithm doesn't help. To be rigorous is very important but needs no specific knowledge (indeed, learning formal logic and coding is a good training for intellectual rigor, to acquire habit to checking every reasoning. But to learn with dibbuk chaverim is perhaps better). There are books particularly good to learn as models of Talmudic study and are not hard to learn. For instance, Shaagat Arieh, Nachalat David, Shut Bet Halevi, Minchat Chinuch. To learn the contents of Talmudic thought, Ktsot Hachoshen is good, it's also relatively easy to read.
To learn seriously Gemara with Rashi and Tosfot, the Maharsha helps a lot to understand the logical nuances in Tosfot.
In summary the steps before entering the data in a program are the core of Talmudic study.