The Talmud On Transitivity
Shlomo Naeh and Uzi Segal
September 3, 2009
Transitivity is a fundamental requirement for consistency. Legalsystems, especially when composed over time and by different agen-cies, may encounter nontransitive cycles. This paper shows that treat-ments of violations of transitivity already appear in the rabbinic lit-erature, starting with the Mishnah and the Talmud (1st5th c CE).This literature offers several solutions that are similar to those used inthe modern economic literature, as well as some other solutions thatmay be adopted in modern situations.
We analyze several examples. One where nontransitive relationsare acceptable; one where a violation of transitivity leads to problemswith extended choice functions; and a third where a nontransitivecycle is deliberately created in order to prevent injustice.
Talmud Department, Hebrew University, and the Shalom Hartman Institute, Israel(firstname.lastname@example.org).
Dept. of Economics, Boston College, Chestnut Hill MA 02467, USA (email@example.com).
Legal system are created over time and by different agencies and it is thereforehardly surprising that occasionally inconsistencies are discovered and needto be resolved. Most systems have some built-in resolution mechanisms, forexample, legislative agencies are ranked as are courts. Such mechanismsare suitable for solving conflicts where two rules contradict each other for example, when laws are declared unconstitutional or when higher courtsnullify decisions made by lower courts.
There is another type of inconsistency, where any pair of two legislationsis consistent, but two laws in conjunction imply a conclusion that is opposedby another law. Such is the case of nontransitive cycles, where by one law Ais ranked above B, by another law B is ranked higher than C, but a third lawranks C higher than A. Such cycles may appear, for example, when rightsof creditors are ranked, or where legislatures must be ranked.
In this paper we describe how the legal system of the Talmud1 copeswith this challenge. Our purpose is twofold. We show that modern attitudesto violations of transitivity have a long history and more importantly, we usethis ancient literature to offer new perspectives, interpretations, and solutionsfor such situations.
There are several theoretical objections to violations of transitivity. Thefirst relates to the requirement that judges use logical reasoning in their de-cisions (see Levi  and Dworkin ). If three rules create a cycle as aboveand a dispute between A and C needs to be resolved, then the logical con-clusion that ranks A higher than C clashes with the direct ruling concerningthese two options, leaving the judge with contradicting instructions.
Transitivity is also a key axiom in social choice theory (see e.g. Harsanyi and Sen ). The best support for transitivity in this context, or in generalchoice situations, comes from the requirement that choice should be consis-tent (see e.g. Sugden  or Rubinstein [?]). Let X be a set of alternatives,and let c be a choice function defined on all nonempty subsets of X. Sup-pose that for three alternatives x, y, and z, the choice function c satisfiesc (x, y) = x, c (y, z) = y, but c (x, z) = z. What should c (x, y, z) be? If theoutcome of c is a singleton (say x), then there is another outcome that iseven better. And if the outcome of c (x, y, z) is not a singleton, then one of
1See Appendix 1 for a glossary and short bios of items marked with a dagger. In thispaper we refer by the term Talmud to the whole corpus of the rabbinic literature.
these outcomes is dominated by the other.2
The Talmud accepts the logic of transitivity and use it in many differentways. For example, since the ritual reading of the book of Esther is soimportant that it suspends temple service, and since temple service is moreimportant than (religious) study, the Talmud [25, Meg. 3a] concludes, in linewith modern reasoning, that schools are canceled for the ritual reading ofthe book of Esther.3
Because of the wide acceptance of transitive reasoning, the Talmud issensitive to violations of this rule. In this paper we discuss three examplesof such violations. The first is a set of three different activities x, y, andz, for which the Rabbis offer reasons why x overrides y, y overrides z, andz overrides x. As this nontransitive cycle does not create any operationalinconsistency, the Talmud ignores the theoretical problem, very much likesome of the solutions offered in the modern economic literature for individualviolations of transitivity (see e.g. Fishburn and LaValle ). We expand onthis in Section 2 below.
In Section 3 we analyze a problem from the Babylonian Talmud wherea choice function defined over two-item sets cannot extend to include three-item sets. The Talmud offers two solutions. One suggests a violation of theindependence of irrelevant alternatives (IIA) axiom. The other is based onranking priority rules rather than options. Solutions that are similar to the
2In the decision theoretic literature there is another argument against violations oftransitivity which is based on Dutch book (or money pump) manipulations (see Ram-sey [18, p. 78], Tversky [29, p. 45], and Yaari [30, p. 36]). Suppose that a decision makerholds x, and prefers x to y, y to z, but z to x. Offer him first to switch from x to z. As heprefers z to x, he will accept. Then offer him to switch from z to y, which once again hewill accept. Finally, for a small commission, offer him to switch from y to x (charge thecommission only at the last step to avoid the issue of income-dependent preferences). Ifpreferences are continuous, he will switch once more, to be back where he started (hold-ing x), minus the commission he paid. Obviously this behavior is foolish why pay fornothing?
3Transitive chains can be used by the Talmud in nonstandard ways. When searchingfor the meaning of a certain biblical phrase, the Rabbis may compare it to an identicalphrase in a different context, claiming that both occurrences must have the same meaning.In several cases they offer a sequence of such citations, each has a common word with itsdirect predecessor. The last verse then determines the meaning of the first one. See forexample [25, Eruv. 51a, Pes. 7b, Ket. 45b,46a], [26, Shab. 2:3, 4d]. Another example fora non-trivial application of transitivity is the ruling that if person A owes B money, andB owes C, then A is considered a debtor of C, with the implication that B cannot waiveAs debt (see [25, Ket. 19a]).
former exist in the economic literature, but the latter solution is new, andas we show, can be formalized into a procedural solution to nontransitivecycles.
Our last case is probably the most interesting it involves medievalinterpretations of a difficult Talmudic text, where legal requirements clashwith considerations for justice. One of the interpretations suggests preservinga nontransitive cycle in order to force conflicting parties into a compromise.We discuss some possible applications of this idea in Section 4.
A few remarks are needed here on the nature of the rabbinic materialsused in this article. Essentially, Talmudic Law encompasses all aspects of[Jewish] life, legal as well as ritual. The Rabbis treat all of them with thesame seriousness and meticulousness. Moreover, they use the same analyticaltools in their discussions of the different materials, be it criminal law, civillaw, temple rituals or other religious commandments, public or individual.This aspect of rabbinic Law characterizes the examples we use in this pa-per. Another important aspect of these materials is their interpretive nature.Typically, the Talmudic passage is built through interpretation and counter-interpretation of previous sources. In this article, we give special attentionto the interpretive principles and processes, explicit or implicit, of the texts.
Some, if not all, of the examples discussed below may seem contrived.Indeed they are. The way of the Talmud is to create such examples in orderto isolate or examine the fine details of an idea or a rule (see Moskovitz ).
Throughout the paper, we use many Hebrew and Aramaic sources. When-ever an English translation exists, we make a reference to it as well. In Ap-pendix 1, we give brief descriptions of the different sources cited and therabbinic terms used in the article.
2 Inconsistent Rules
The Law does not cover all possible cases, and courts may need to extrap-olate from existing laws. This is certainly the case with casuistic Law, likethe Talmudic Law. Transitive deduction is a simple, yet powerful tool forsuch extrapolations and is indeed used in the rabbinic literature. But whathappens when such deductions contradict other rulings or deductions?
Such situations create two types of difficulties. One is philosophical we expect a legal system to be free of logical contradictions. The other oneis practical which logical deduction should we follow? It is important to
observe that although in many cases cyclic rules need to be resolved becausethey lead to contradictory rulings, there are situation where such rules maycoexist. Although they form a philosophical or a logical problem, they donot lead to any operational conflicts. As we show in this section, the Talmudis not bothered by such cycles.
Consider the following observations which are based on the analysis ofbiblical verses.
1. The laws of Sabbath are suspended in order to perform temple services.Such services are performed on the Sabbath even though the JewishLaw forbids many of these acts (e.g. killing animals) during that day.
2. Executions do not suspend Sabbath.4
The Talmud deliberates: If Sa