Fuzzy measurement in the Mishnah and the Talmud

Artificial Intelligence and Law

Volumn 7, Issue 2, 1999, Pages 273-288

  Shapira, Ron A ^a  

^a TEL AVIV UNIVERSITY   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

FUZZY SETS; MATHEMATICAL MODELS; UNCERTAIN SYSTEMS;

FUZZY MEASUREMENT;

ARTIFICIAL INTELLIGENCE;

EID: 0033362442     PISSN: 09248463     EISSN: None     Source Type: Journal    
DOI: 10.1023/a:1008321725690     Document Type: Article

References (70)

1. Menachem Fisch, To Know Wisdom: Science, Rationality and Torah-Study (1994), pp. 41-98.
2. Suzanne Last Stone, "In Pursuit of the Counter-Text: The Turn to the Jewish Legal Model in Contemporary American Legal Theory", 106 Harv. L. Rev. 813 (1993).
3. See: Elimelech Westreich, "Medicine and Natural Sciences in Judgments of Rabbinical Courts", 26 Mishpatim (Hebrew University Law Review) 425 (1996) [Hebrew].
4. Mishnah, Tractate Bekhoroth [Firstborns], 4:4.
5. See:: Exodus 22:30. Note that common English translations of this Biblical verse are inaccurate.
6. Westreich, supra note 4, at p. 418, correctly notes that this early record of talmudic law accepting scientific evidence failed to draw adequate attention.
7. Tosephta, Tractate Nidah [Menstruous woman] 4:3. A later version of the same report is mentioned in the Babylonian Talmud, Tractate Nidah [Menstruous woman] 22b. The Talmud adds that the sages finally concluded the case by conducting an experiment: they immersed the "skin patches" in water, and decided that, if they melt, the woman would be declared menstruous.
8. Leviticus 15:19; 18:19.
9. Leviticus 15:19; 18:19.
10. Bas C. van Fraassen, Laws and Symmetry (1989) 2-7.
11. Babylonian Talmud, Tractate Sanhedrin [Synod], 11b.
12. One should not get the impression that ancient Jewish law (or the author of this paper) are obsessed with female organs. The common denominator shared by these recurring examples is completely accidental.
13. Babylonian Talmud, Tractate Ketubboth [Marriage Contracts], 11b.
14. Jerusalem Talmud, Tractate Sanhedrin [Synod], ch. 1:2; see also: Jerusalem Talmud, Tractate Nedarim [Vows], 6:8; Rabbi Avin in: Jerusalem Talmud, Tractate Ketubboth [Marriage Contracts] 1:2.
15. Jerusalem Talmud, Tractate Sanhedrin [Synod], ch. 1:2; see also: Jerusalem Talmud, Tractate Nedarim [Vows], 6:8; Rabbi Avin in: Jerusalem Talmud, Tractate Ketubboth [Marriage Contracts] 1:2.
16. Jerusalem Talmud, Tractate Sanhedrin [Synod], ch. 1:2; see also: Jerusalem Talmud, Tractate Nedarim [Vows], 6:8; Rabbi Avin in: Jerusalem Talmud, Tractate Ketubboth [Marriage Contracts] 1:2.
17. See par. A. iv. below.
18. As has been observed by Ana anonymous referee of this paper, this conclusion might entail such normative absurdities as the retroactive exoneration of a convicted rapist.
19. Tofaphoth, Babylonian Talmud, Tractate Nidah Menstruous Woman], 64b. the length of pregnancy.
20. 18 and the appearance of male puberty signs.
21. This conclusion was perfectly compatible with the prevalent world view during the talmudic era, which was occasionalism (also known as voluntarism), namely, the view that everything happens because it is directly and individually willed by God. In late antiquity, this was a well-established doctrine in Judaism.
22. 107b: "The Holy One sustains all, from buffalo's horns to nits".
23. See Matthew 10:30: "But the very hairs of your head are all numbered".
24. Guide of the perplexed, 3:17.
25. Summa Contra Gentiles, bk. 2, sec. 4-9.
26. See the dispute about the ritual impurity of "the oven of Akhnai" in Mishnah, tractate Kelim [Tools] 5:10; Mishnah, Tractate Eduyot [Testimonies] 7:7.
27. See the dispute about the ritual impurity of "the oven of Akhnai" in Mishnah, tractate Kelim [Tools] 5:10; Mishnah, Tractate Eduyot [Testimonies] 7:7.
28. Babylonian Talmud, Tractate Rosh Hashanah [New Year's Day], 13a.
29. Ibid.
30. Ibid.
31. M. Silberg, Principia Talmudica 45-59 (1961).
32. Mishnah, Tractate Bava Batra [Last Section], chapter B, rule 6.
33. Babylonian Talmud, Tractate Bava Batra [Last Section] 23b.
34. RSI, ibid.
35. Alex Klein, "Rabbi Yrmiya's Questions", 3 Higayon 151 (1996) [Hebrew]. For other reflections of the conflict between Rabbi Yrmiya and the majority of sages see: Babylonian Talmud, Tractate Rosh Hashanah [New Year's Day] 13a; Babylonian Talmud, Tractate Sotah [Faithless Wife] 16b.
36. Alex Klein, "Rabbi Yrmiya's Questions", 3 Higayon 151 (1996) [Hebrew]. For other reflections of the conflict between Rabbi Yrmiya and the majority of sages see: Babylonian Talmud, Tractate Rosh Hashanah [New Year's Day] 13a; Babylonian Talmud, Tractate Sotah [Faithless Wife] 16b.
37. Alex Klein, "Rabbi Yrmiya's Questions", 3 Higayon 151 (1996) [Hebrew]. For other reflections of the conflict between Rabbi Yrmiya and the majority of sages see: Babylonian Talmud, Tractate Rosh Hashanah [New Year's Day] 13a; Babylonian Talmud, Tractate Sotah [Faithless Wife] 16b.
38. A. Kirschenbaum, Equity in Jewish Law, vol. I, p. 5 (1991).
39. Silberg, supra note 28, at p. 52.
40. See: Russell Christopher "Unknowing Justification and the Logical Necessity of the Dadson Principle in Self-Defence", 15 Oxford J. Legal Stud. 229 (1995).
41. Other well-known examples: (a) On her first delivery, a ewe gives birth to two lambs whose heads come out at the same time. The biblical law states that "the first born male" is to be sacrificed as a priestly offering (Exodus 13:12), and a dispute arises as to whether there could be two "firstborns" ( Mishnah, Tractate Bechoroth [Firstborns] 17:1). (b) Simultaneous testimonies (Babylonian Talmud, Tractate Shvu'oth [Oaths] 32a). Half-broken partition. A "partition" or fence mark the boundary of a "domain". The precise scope of this domain has important implications concerning property law, the law of defilement, the law of Shabbath, and other legal fields. A rule prescribes that, to determine the scope of the "domains", the broken section of the fence must not be larger than the unimpaired section. The possibility of a partition wherein the broken and unbroken sections are exactly equal is discussed and debated (Babylonian Talmud, Tractate Eruvin [Amalgamation of Courts] 5; 15b).
42. Other well-known examples: (a) On her first delivery, a ewe gives birth to two lambs whose heads come out at the same time. The biblical law states that "the first born male" is to be sacrificed as a priestly offering (Exodus 13:12), and a dispute arises as to whether there could be two "firstborns" ( Mishnah, Tractate Bechoroth [Firstborns] 17:1). (b) Simultaneous testimonies (Babylonian Talmud, Tractate Shvu'oth [Oaths] 32a). Half-broken partition. A "partition" or fence mark the boundary of a "domain". The precise scope of this domain has important implications concerning property law, the law of defilement, the law of Shabbath, and other legal fields. A rule prescribes that, to determine the scope of the "domains", the broken section of the fence must not be larger than the unimpaired section. The possibility of a partition wherein the broken and unbroken sections are exactly equal is discussed and debated (Babylonian Talmud, Tractate Eruvin [Amalgamation of Courts] 5; 15b).
43. Other well-known examples: (a) On her first delivery, a ewe gives birth to two lambs whose heads come out at the same time. The biblical law states that "the first born male" is to be sacrificed as a priestly offering (Exodus 13:12), and a dispute arises as to whether there could be two "firstborns" ( Mishnah, Tractate Bechoroth [Firstborns] 17:1). (b) Simultaneous testimonies (Babylonian Talmud, Tractate Shvu'oth [Oaths] 32a). Half-broken partition. A "partition" or fence mark the boundary of a "domain". The precise scope of this domain has important implications concerning property law, the law of defilement, the law of Shabbath, and other legal fields. A rule prescribes that, to determine the scope of the "domains", the broken section of the fence must not be larger than the unimpaired section. The possibility of a partition wherein the broken and unbroken sections are exactly equal is discussed and debated (Babylonian Talmud, Tractate Eruvin [Amalgamation of Courts] 5; 15b).
44. Other well-known examples: (a) On her first delivery, a ewe gives birth to two lambs whose heads come out at the same time. The biblical law states that "the first born male" is to be sacrificed as a priestly offering (Exodus 13:12), and a dispute arises as to whether there could be two "firstborns" ( Mishnah, Tractate Bechoroth [Firstborns] 17:1). (b) Simultaneous testimonies (Babylonian Talmud, Tractate Shvu'oth [Oaths] 32a). Half-broken partition. A "partition" or fence mark the boundary of a "domain". The precise scope of this domain has important implications concerning property law, the law of defilement, the law of Shabbath, and other legal fields. A rule prescribes that, to determine the scope of the "domains", the broken section of the fence must not be larger than the unimpaired section. The possibility of a partition wherein the broken and unbroken sections are exactly equal is discussed and debated (Babylonian Talmud, Tractate Eruvin [Amalgamation of Courts] 5; 15b).
45. See: RASHI (Rabbi Shlomo Itzhaki), Commentaries on Babylonian Talmud, Tractate Bechoroth [Firstborns] 13a: a prohibition on performing levirate, unless the man and the widow show that they were exclusively motivated by true religious purposes (11th cent). Cf.: Rabbi Moshe Ben Meimon, Misneh Torah, Rules of Levirate and Halizah, 1:2.
46. See: RASHI (Rabbi Shlomo Itzhaki), Commentaries on Babylonian Talmud, Tractate Bechoroth [Firstborns] 13a: a prohibition on performing levirate, unless the man and the widow show that they were exclusively motivated by true religious purposes (11th cent). Cf.: Rabbi Moshe Ben Meimon, Misneh Torah, Rules of Levirate and Halizah, 1:2.
47. Mishnah, Tractate Yevamoth [Levirate Marriages], 1:1.
48. According to biblical law a man was allowed to marry more than one woman, and polygamy was abolished only in the 11th century.
49. Leviticus 18:18: "Neither shalt thou take a wife to her sister, to vex her, to uncover her nakedness, beside the other in her life time".
50. Babylonian Talmud, Tractate Yevamoth [Levirate Marriages], 19a.
51. P. J. Green and R. Sibson, "Computing Dirichlet Tessellations in the Plane", 21 Computer J. 168 (1978).
52. Babylonian Talmud, Tractate Bava Batra [Last Section] 23b: "majority and vicinity - majority overrides".
53. See several proposed algorithms in: E. Merzbach, B. Singer "The Nearest Town in the Law of 'Beheaded Heifer'", 2 Higayon 76 (1993) [Hebrew].
54. See: Tosephta, Tractate Eruvin [Amalgamation of Courts] 6a; Babylonian Talmud, Tractate Bechoroth [Firstborns] 17a.
55. See: Tosephta, Tractate Eruvin [Amalgamation of Courts] 6a; Babylonian Talmud, Tractate Bechoroth [Firstborns] 17a.
56. See: Tosephta, Tractate Hullin [Profane Slaughtering] 28b.
57. Ibid. See supra note 20 and accompanying text (concerning occasionalism).
58. Ibid. See supra note 20 and accompanying text (concerning occasionalism).
59. Babylonian Talmud, Tractate Bechoroth [Firstborns] 18a.
60. Ibid.
61. 2, ∞). See: D. Dubois and H. Prade, "Fuzzy Real Algebra" 2 Fuzzy Sets and Systems 327 (1979); D. Dubois, H. Prade, Possibility Theory 48-50 (1988).
62. 2, ∞). See: D. Dubois and H. Prade, "Fuzzy Real Algebra" 2 Fuzzy Sets and Systems 327 (1979); D. Dubois, H. Prade, Possibility Theory 48-50 (1988).
63. Notoriously, fuzzy equations behave differently from crisp equations. A standard way of defining arithmetic operations on fuzzy numbers is by operations on their α-cuts (see infra) according to the well-established principles of interval analysis. However, according to such a definition we can show, for example, that B - A is not be the solution for A + x = B.
64. Z. Pawlak, "Rough Sets", 17 Fuzzy Sets and Systems 99 (1985).
65. Z. Pawlak, S. K. M. Wong, and W. Ziarko, "Rough Sets: Probabilistic Versus Deterministic Approach", 29 Int. J. Man-Mach. Stud. 81 (1988).
66. αF, is a crisp set that contains all the elements of the universe whose membership values in F are greater than or equal to the specified value α (which is a member of the range of F).
67. 55 The opposition may well be reconstructed as claiming that even a core-less intersection may suffice in some cases for the legal purpose.
68. For the special problematics of such equations see George J. Klir and Bo Yuan, Fuzzy Sets and Fuzzy Systems: Theory and Application 114 et seq. (1995).
69. See: Klir and Yuan, ibid., at pp. 250-1.
70. α A) is the measure of the α-cut of A defined by the Lebesge integral of its characteristic function.