An introduction to alysidal algebra (IV)

Purpose – Deontical impure systems are systems whose object set is formed by an s-impure set, whose elements are perceptuales significances (relative beings) of material and/or energetic objects (absolute beings) and whose relational set is freeways of relations, formed by sheaves of relations going in two-way directions and at least one of its relations has deontical properties such as permission, prohibition, obligation and faculty. The paper aims to discuss these issues. Design/methodology/approach – Mathematical and logical development of human society ethical and normative structure. Findings – Existence of relations with positive imperative modality (obligation) would constitute the skeleton of the system. Negative imperative modality (prohibition) would be the immunological system of protection of the system. Modality permission the muscular system, that gives the necessary flexibility. Four theorems have been formulated based on Gödel's theorem demonstrating the inconsistency or incompleteness of DISs. For each constructed systemic conception can happen to it one of the two following things: either some allowed responses are not produced or else some forbidden responses are produced. Responses prohibited by the system are defined as nonwished effects. Originality/value – This paper is a continuation of the four previous papers and is developed the theory of deontical impure systems.

Reality, Systems and Impure Systems

Impure systems contain Objects and Subjects: Subjects are human beings. We can distinguish a person as an observer (subjectively outside the system) and that by definition is the Subject himself, and part of the system. In this case he acquires the category of object. Objects (relative beings) are significances, which are the consequence of perceptual beliefs on the part of the Subject about material or energetic objects (absolute beings) with certain characteristics.The IS (Impure System) approach is as follows: Objects are perceptual significances (relative beings) of material or energetic objects (absolute beings). The set of these objects will form an impure set of the first order. The existing relations between these relative objects will be of two classes: transactions of matter and/or energy and inferential relations. Transactions can have alethic modality: necessity, possibility, impossibility and contingency. Ontic existence of possibility entails that inferential relations have Deontic modality: obligation, permission, prohibition, faculty and analogy. We distinguished between theorems (natural laws) and norms (ethical, legislative and customary rules of conduct).

Los derechos de los vencidos: las capitulaciones de Granada (1491)

El artículo propone una lectura de las cláusulas religiosas de las capitulaciones para la entrega de Granada (25 de noviembre de 1491). Se pone en tela de juicio la idea, comúnmente aceptada en la historiografía, de que las capitulaciones establecían el respeto a las leyes y a la religión de los vencidos y que, por lo tanto, las actuaciones que llevaron a la conversión de los musulmanes de Granada en los años 1499-1501 violaron las mismas capitulaciones. La comparación entre las peticiones del rey Boabdil y el texto firmado por los Reyes Católicos, apoyada sobre el derecho canónico, permite aclarar la posición de los Reyes en puntos tan decisivos como las modalidades del culto islámico, las señales de identificación, la prohibición de las conversiones forzadas y el estatuto de los hombres y mujeres convertidos del Cristianismo al Islam (elches). Se intenta demostrar que las capitulaciones, lejos de fijar la situación religiosa en Granada, establecían la dominación cristiana, abriendo el paso a una dinámica de conversión de la población musulmana y a la recuperación de los elches por la Iglesia.

“Unintended effects”: A theorem for complex systems

Unintended effects are well known to economists and sociologists and their consequences may be devastating. The main objective of this article is to formulate a mathematical theorem, based on Gödel's famous incompleteness theorem, in which it is shown, that from the moment deontical modalities (prohibition, obligation, permission, and faculty) are introduced into the social system, responses are allowed by the system that are not produced, however, prohibited responses or unintended effects may occur.