Fuzzy measurement in the Mishnah and the Talmud

Abstract

I discuss the attitude of Jewish law sources from the 2nd---:5th centuries to the imprecision of measurement. I review a problem that the Talmud refers to, somewhat obscurely, as "impossible reduction". This problem arises when a legal rule specifies an object by referring to a maximized (or minimized) measurement function, e.g., when a rule applies to "the largest part" of a divided whole, or to "the first" incidence that occurs, etc. A problem that is often mentioned is whether there might be hypothetical situations involving more than one maximal (or minimal) value of the relevant measurement and, given such situations, what is the pertinent legal rule. Presumption of simultaneous occurrences or equally measured values are also a source of embarrassment to modern legal systems, in situations exemplified in the paper, where law determines a preference based on measured values. I contend that the Talmudic sources discussing the problem of "impossible reduction" were guided by primitive insights compatible with fuzzy logic presentation of the inevitable uncertainty involved in measurement. I maintain that fuzzy models of data are compatible with a positivistic epistemology, which refuses to assume any precision in the extra-conscious "world" that may not be captured by observation and measurement. I therefore propose this view as the preferred interpretation of the Talmudic notion of "impossible reduction". Attributing a fuzzy world view to the Talmudic authorities is meant not only to increase our understanding of the Talmud but, in so doing, also to demonstrate that fuzzy notions are entrenched in our practical reasoning. If Talmudic sages did indeed conceive the results of measurements in terms of fuzzy numbers, then equality between the results of measurements had to be more complicated than crisp equations. The problem of "impossible reduction" could lie in fuzzy sets with an empty core or whose membership functions were only partly congruent. "Reduction is impossible" may thus be reconstructed as "there is no core to the intersection of two measures". I describe Dirichlet maps for fuzzy measurements of distance as a rough partition of the universe, where for any region A there may be a non-empty set of ăÂ? - _A (upper approximation minus lower approximation), where the problem of "impossible reduction" applies. This model may easily be combined with probabilistic extention. The possibility of adopting practical decision standards based on Â?-cuts (and therefore applying interval analysis to fuzzy equations) is discussed in this context. I propose to characterize the uncertainty that was presumably capped by the old sages as "U-uncertainty", defined, for a non-empty fuzzy set A on the set of real numbers, whose Â?-cuts are intervals of real numbers, as U(A) = 1/h(A) Â? 0 h(A) log [1+μ(Â?A)]dÂ?, where h(A) is the largest membership value obtained by any element of A and μ(Â?A) is the measure of the Â?-cut of A defined by the Lebesge integral of its characteristic function.