The Talmud On Transitivity Shlomo Naeh∗ and Uzi Segal† September 3, 2009 Abstract Transitivity is a fundamental requirement for consistency. Legal systems, 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 (1st–5th c CE). This literature offers several solutions that are similar to those used in the modern economic literature, as well as some other solutions that may be adopted in modern situations. We analyze several examples. One where nontransitive relations are acceptable; one where a violation of transitivity leads to problems with extended choice functions; and a third where a nontransitive cycle is deliberately created in order to prevent injustice. ∗Talmud Department, Hebrew University, and the Shalom Hartman Institute, Israel (
[email protected]). †Dept. of Economics, Boston College, Chestnut Hill MA 02467, USA (
[email protected]). 1 Introduction Legal system are created over time and by different agencies and it is therefore hardly surprising that occasionally inconsistencies are discovered and need to be resolved. Most systems have some built-in resolution mechanisms, for example, legislative agencies are ranked as are courts. Such mechanisms are suitable for solving conflicts where two rules contradict each other — for example, when laws are declared unconstitutional or when higher courts nullify decisions made by lower courts. There is another type of inconsistency, where any pair of two legislations is consistent, but two laws in conjunction imply a conclusion that is opposed by another law.