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• Logic and Set Theory
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## Abstract

The logical reasoning first appeared within the Babylonian legal tradition established by the Sumerians in the law codes which were first over the world: Ur-Nammu (ca. 2047 – 2030 B.C.); Lipit-Ishtar (ca. 1900 – 1850 B.C.), and later by their successors, the Akkadians: Hammurabi (1728 – 1686 B.C.). In these codes the casuistic law formulation began first to be used: “If/when (Akkadian: šumma) this or that occurs, this or that must be done” allowed the Akkadians to build up a theory of logical connectives: “... or…”, “… and…”, “if…, then…”, “not…” that must have been applied in their jurisprudence. So, a trial decision looked like an inference by modus pones and modus tollens or by other logical rules from (i) some facts and (ii) an appropriate article in the law code represented by an ever true implication. The law code was announced by erecting a stele with the code or by engraving the code on a stone wall. It was considered a set of axioms announced for all. Then the trial decisions are regarded as claims logically inferred from the law code on the stones. The only law code of the Greeks that was excavated is the Code of Gortyn (Crete, the 5th century B.C.). It is so similar to the Babylonian codes by its law formulations; therefore, we can suppose that the Greeks developed their codes under a direct influence of the Semitic legal tradition: the code was represented as the words of the stele and the court was a logic application from these words. In this way the Greek logic was established within a Babylonian legal tradition, as well. Hence, we can conclude that, first, logic appeared in Babylonia and, second, it appeared within a unique legal tradition where all trial decisions must have been transparent, obvious, and provable. The symbolic logic appeared first not in Greece, but in Mesopotamia and this tradition was grounded in the Sumerian/Akkadian jurisprudence.

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## Abstract

Hezekiah bar Halafta and Judah Messer Leon, who wrote in 14th – 15th century in Provence and Italy, were the first and last of “Jewish Schoolman.” This short article compares two texts, in order to showing differences and similarities.

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## Abstract

This paper tries to outline a history of development of informal logic in Semitic languages and especially in Arabic. It tries to explain how the first definite formulation of rules of this logic appeared at āl-Šāfi‘y’s Risāla, a work on ’uswl āl-fiqh or methodology of law. It attempts also to provide new theories and hypotheses about the translation movement in the Arabic and Islamic medieval world.

Open access

## Abstract

A lively exchange in recent epistemology considers the problem of epistemic disagreement between peers: disagreement between those who share evidence and have equal cognitive abilities. Two main views have emerged about how to proceed in such circumstances: be steadfast in maintaining one’s own view or conciliate, and suspend or reduce one’s confidence in one’s belief. Talmudic debates do seem to promote steadfastness, as the disputants are not called on to conciliate purely because they confront a disagreeing peer. But why? Third party judgments are even more problematic, for what epistemic warrant is there for choosing between a disagreement of superiors? A common explanation for Talmudic steadfastness is the notion ’elu w’elu divrey ’Elohim kayim – both sides of Talmudic (or, more generally, halakhic) disputes have ‘heavenly’ legitimacy. But a closer look at this oft-quoted dictum and its various interpretations does not, in fact, reveal such support for steadfastness. Other explanations for Talmudic steadfastness are, therefore, required.

Open access

## Abstract

The purpose of this study is to show that the logical content of a Tann’ayitic hermeneutic changed and developed as it passed into the hands of the ’Amor’ayim, the Tann’ayim’s successors, and then into the anonymous stratum of the Babylonian Talmud. This hermeneutic was based on a very specific syntactical order in a biblical verse, which was formed by an initial inclusive clause, followed by a list of specifics, and then followed by a second inclusive clause. This hermeneutic is called in Hebrew kelal uferaṭ ukelal. In the Tann’ayitic period the hermeneutic required that the second inclusive clause had to be more extensive than the first one. It appears that this new degree of extensiveness suggested that the list of specifics was not definitive of the initial inclusive clause and that other things might be implied by the second one. The way that the rabbinic interpreter determined what these things might be was by seeking the common characteristics that the items in the specifics clause shared. By the time of the late Tann’ayim and early ’Amor’ayim the requirement for the two inclusive clauses had changed. The formal syntax of the hermeneutic remained, but inclusive clauses had to be equal in their degree of inclusivity. The change in logic seems to be the result of viewing a second, more inclusive clause as a distinct element that could be disconnected from the first inclusive clause and the specifics that follow it. If the two inclusive clauses were, however, the same or similar, the rabbinic interpreter could argue that they belonged to the same categories and thus formed a legitimate kelal uferaṭ ukelal. In the final period of the Talmud’s creation neither the syntactic nor logical requirements were any longer needed to form a kelal uferaṭ ukelal. Two artificially constructed inclusive clauses and some specifics could appear in almost any order within a biblical verse and be considered a kelal uferaṭ ukelal. It appears that the desire of the rabbinic interpreters of each era to connect their halakot to the Torah was the force behind the changes we have described.

Open access

## Abstract

Gravestones with Hebrew inscriptions are the most common class of Jewish monuments still present in such regions as Ukraine or Belarus. Epitaphs are related to various Biblical, Rabbinical, and liturgical texts. Despite that, the genre of Hebrew epitaphs seldom becomes an object of cultural or literary studies. In this paper, I show that a function of Hebrew epitaphs is to connect the ideal world of Hebrew sacred texts to the world of everyday life of a Jewish community. This is achieved at several levels. First, the necessary elements of an epitaph – name, date, and location marker – place the deceased person into a specific absolute context. Second, the epitaphs quote Biblical verses with the name of the person thus stressing his/her similarity to a Biblical character. Third, there is Hebrew/Yiddish orthography code-switching between the concepts found in the sacred books and those from the everyday world. Fourth, the epitaphs occupy an intermediate position between the professional and folk literature. Fifth, the epitaphs are also in between the canonical and folk religion. I analyze complex hermeneutic mechanisms of indirect quotations in the epitaphs and show that the methods of actualization of the sacred texts are similar to those of the Rabbinical literature. Furthermore, the dichotomy between the sacred and profane in the epitaphs is based upon the Rabbinical concept of the ‘Internal Jewish Bilingualism’ (Hebrew/Aramaic or Hebrew/Yiddish), which is parallel to the juxtaposition of the Written and Oral Torah.

Open access

## Abstract

In this paper, we study the generalized fractional operators pertaining to the generalized Mittag-Leffler function and multi-index Mittag-Leffler function. Some applications of the established results associated with generalized Wright function are also deduced as corollaries. The results are useful in solving the problems of science, engineering and technology where the Mittag-Leffler function occurs naturally.

Open access

## Abstract

In this study, a numerical solution of the modified Burgers’ equation is obtained by the finite difference methods. For the solution process, two linearization techniques have been applied to get over the non-linear term existing in the equation. Then, some comparisons have been made between the obtained results and those available in the literature. Furthermore, the error norms L2 and L are computed and found to be sufficiently small and compatible with others in the literature. The stability analysis of the linearized finite difference equations obtained by two different linearization techniques has been separately conducted via Fourier stability analysis method.

Open access

## Abstract

Based on progressively Type-II censored samples, this paper deals with the estimation of R = P(X < Y) when X and Y come from two independent inverted exponentiated rayleigh distributions with different shape parameters, but having the same scale parameter. The maximum likelihood estimator and UMVUE of R is obtained. Different confidence intervals are presented. The Bayes estimator of R and the corresponding credible interval using the Gibbs sampling technique are also proposed. Monte Carlo simulations are performed to compare the performances of the different methods. One illustrative example is provided to demonstrate the application of the proposed method.

Open access

## Abstract

Several versions of the Stein-rule estimators of the coefficient vector in a linear regression model are proposed in the literature. In the present paper, we propose new feasible generalized Stein-rule restricted ridge regression estimators to examine multicollinearity and autocorrelation problems simultaneously for the general linear regression model, when certain additional exact restrictions are placed on these coefficients. Moreover, a Monte Carlo simulation experiment is performed to investigate the performance of the proposed estimator over the others.