Talmudic reasoning

There are principles of Talmudic hermeneutics which constitute an interesting method alternative, although related in some instances, to the syllogism. Three groups are listed: the 7 rules of Hillel, the 13 rules of R. Ishmael, and the 32 rules of R. Eliezer b. Jose Ha-Gelili (all come from the Jewish Encyclopedia 1906).

7 rules of Hillel 

1. Ḳal (ḳol) wa-ḥomer (קל וחומר): "Argumentum a minori ad majus" or "a majori ad minus"; corresponding to the scholastic proof a fortiori.
2. Gezerah shawah (גזירה שוה): Argument from analogy. Biblical passages containing synonyms or homonyms are subject, however much they differ in other respects, to identical definitions and applications.
3. Binyan ab mi-katub eḥad (בנין אב מכתוב אחד): Application of a provision found in one passage only to passages which are related to the first in content but do not contain the provision in question.
4. Binyan ab mi-shene ketubim (בנין אב מכתוב אחד): The same as the preceding, except that the provision is generalized from two Biblical passages.
5. Kelal u-Peraṭ and Peraṭ u-kelal (כלל ופרט ופרט וכלל): Definition of the general by the particular, and of the particular by the general.
6. Ka-yoẓe bo mi-maḳom aḥer (כיוצא בו ממקום אחר): Similarity in content to another Scriptural passage.
7. Dabar ha-lamed me-'inyano (דבר הלמד מעניינו): Interpretation deduced from the context.

13 rules of R. Ishmael b. Elisha: (from Jewish Encyclopedia 1906)

1. Ḳal wa-ḥomer: Identical with the first rule of Hillel.
2. Gezerah shawah: Identical with the second rule of Hillel.
3. Binyan ab: Rules deduced from a single passage of Scripture and rules deduced from two passages. This rule is a combination of the third and fourth rules of Hillel.
4. Kelal u-Peraṭ: The general and the particular.
5. u-Peraṭ u-kelal: The particular and the general.
6. Kelal u-Peraṭ u-kelal: The general, the particular, and the general.
7. The general which requires elucidation by the particular, and the particular which requires elucidation by the general.
8. The particular implied in the general and excepted from it for pedagogic purposes elucidates the general as well as the particular.
9. The particular implied in the general and excepted from it on account of the special regulation which corresponds in concept to the general, is thus isolated to decrease rather than to increase the rigidity of its application.
10. The particular implied in the general and excepted from it on account of some other special regulation which does not correspond in concept to the general, is thus isolated either to decrease or to increase the rigidity of its application.
11. The particular implied in the general and excepted from it on account of a new and reversed decision can be referred to the general only in case the passage under consideration makes an explicit reference to it.
12. Deduction from the context.
13. When two Biblical passages contradict each other the contradiction in question must be solved by reference to a third passage.

Rules seven to eleven are formed by a subdivision of the fifth rule of Hillel; rule twelve corresponds to the seventh rule of Hillel, but is amplified in certain particulars; rule thirteen does not occur in Hillel, while, on the other hand, the sixth rule of Hillel is omitted by Ishmael.

32 Rules of Eliezer b. Jose Ha-Gelili

1. Ribbuy (extension): The particles 'et," "gam," and "af," which are superfluous, indicate that something which is not explicitly stated must be regarded as included in the passage under consideration, or that some teaching is implied thereby.
2. Mi'uṭ (limitation): The particles "ak," "raḳ" and "min" indicate that something implied by the concept under consideration must be excluded in a specific case.
3. Ribbuy aḥar ribbuy (extension after extension): When one extension follows another it indicates that more must be regarded as implied.
4. Mi'uṭ aḥar mi'uṭ (limitation after limitation): A double limitation indicates that more is to be omitted.
5. Ḳal wa-ḥomer meforash: "Argumentum a minori ad majus," or vice versa, and expressly so characterized in the text.
6. 6. Ḳal wa-ḥomer satum: "Argumentum a minori ad majus," or vice versa, but only implied, not explicitly declared to be one in the text. This and the preceding rule are contained in the Rules of Hillel, No. 1. 
7. identical with Rule 2 of Hillel.
8. identical with Rule 3 of Hillel 
9. Derek ḳeẓarah: Abbreviation is sometimes used in the text when the subject of discussion is self-explanatory.
10. Dabar shehu shanuy (repeated expression): Repetition implies a special meaning.
11. Siddur she-neḥlaḳ: Where in the text a clause or sentence not logically divisible is divided by the punctuation, the proper order and the division of the verses must be restored according to the logical connection.
12. Anything introduced as a comparison to illustrate and explain something else, itself receives in this way a better explanation and elucidation.
13. When the general is followed by the particular, the latter is specific to the former and merely defines it more exactly (comp. Rules of Hillel, No. 5).
14. Something important is compared with something unimportant to elucidate it and render it more readily intelligible.
15. Same as Rule 13 of R. Ishmael.
16. Dabar meyuḥad bi-meḳomo: An expression which occurs in only one passage can be explained only by the context. This must have been the original meaning of the rule, although another explanation is given in the examples cited in the baraita.
17. A point which is not clearly explained in the main passage may be better elucidated in another passage.
18. A statement with regard to a part may imply the whole.
19. A statement concerning one thing may hold good with regard to another as well.
20. A statement concerning one thing may apply only to something else.
21. If one object is compared to two other objects, the best part of both the latter forms the tertium quid of comparison.
22. A passage may be supplemented and explained by a parallel passage.
23. A passage serves to elucidate and supplement its parallel passage.
24. When the specific implied in the general is especially excepted from the general, it serves to emphasize some property characterizing the specific.
25. The specific implied in the general is frequently excepted from the general to elucidate some other specific property, and to develop some special teaching concerning it.
26. Mashal (parable).
27. Mi-ma'al: Interpretation through the preceding.
28. Mi-neged: Interpretation through the opposite.
29. Gemaṭria: Interpretation according to the numerical value of the letters.
30. Noṭariḳon: Interpretation by dividing a word into two or more parts.
31. Postposition of the precedent. Many phrases which follow must be regarded as properly preceding, and must be interpreted accordingly in exegesis.
32. Many portions of the Bible refer to an earlier period than do the sections which precede them, and vice versa.

References:

• 7 Rules of Hillel: here
• 13 Rules of R. Ishmael: here
• 32 Rules of R. Eliezer: here