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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. Objects and freeways form chains.

The approach used was mathematical and logical development of human society structure.

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 to the system, in as much to the modality faculty its neurocerebral system, because it allows one to make decisions. Transactions of energy, money, merchandise, population, etc. would be the equivalent one to the sanguineous system. These economic transactions and inferential relations, depend, as well, on the existence of a legislative body with their obligations, prohibitions and permissions that regulate them.

This paper is a continuation of Part I, published in Kybernetes, Volume 41, Issue 1/2, 2012, continuing the development of Alysidal Algebra, which is important for the study of deontical impure systems. They are defined coupling functions and alysidal structures. It is defined as a special coupling function denominated gnorpsic function that can be used for algebraic operations between alysidal sets.

The purpose of this paper is to present a further extension of the regular grammars of the language L(M_T).

This language may represent an ecological model, written in a formal mathematical structure.

The paper presents and develops recognoscitive and generative grammars of sentences (flow equations) together with a synonymy relationship, which characterize the L(M_T) language.

Provides a discussion of modelling of language.

The purpose of this paper is to provide an introduction to alysidal algebra. 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. Objects and freeways form chains.

The paper looks at the mathematical and logical development of human society structure.

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 to the system, in as much to the modality faculty its neurocerebral system, because it allows him to make decisions. Transactions of energy, money, merchandise, population, etc. would be the equivalent one to the sanguineous system. These economic transactions and inferential relations, depend, as well, on the existence of a legislative body with their obligations, prohibitions and permissions that regulate them.

The concepts of alysidal set are introduced, whose elements are chains and coupling function between alysidal sets. Environment is formed as well by different DIS. That is to say, by an alysidal set whose elements are simultaneously systems. Specific interchanges (stimulus‐response) leave certain nodes and act on certain nodes of the alysidal sets (stimulus environment‐DIS‐response environment). The paper defines a special coupling function, denominated gnorpsic function, that can be used for algebraic operations between alysidal sets.

All models have in common that they encode experience and always involve signs, signals, syntax, semantics and an ability to decode and derive meaning from what is encoded. Therefore, an ecological model can be considered a text T, written in a formal (mathematical) language defined by the authors as L(M_T).

The language chosen is a regular language generated by regular grammars which in turn are generated by finite automaton.

A text results from a union of some finite automata defining a textual grammar. Recognoscitive and generative grammars of words, defining the basis of the L(M_T) language are developed.

A mathematical modelling of ecological systems is developed in parts I and II of this research project.

A language for ecological systems modelling has been defined and its implementation discussed.

This paper attempts to compare two biological successions, in general any two successions, analysing the differences between them.

An algorithm was designed based on a metric which enables one to calculate the associated distance between two successions and, based on the distance obtained, the type of mutations which have occurred was approached.

The empirical analysis shows that the transformations caused in the successions have been detected by the metric. Today there are numerous enormously powerful programs able to trawl entire databases and therefore the aim of this paper was not to compare these, but to demonstrate a different way of comparing two given successions.

The paper presents a comparison of two DNA successions having a measure of the degree of similarity between them and their possible mutations.

The metric presented is a generalisation of the Hamming distance in strings of different lengths. The software associated with the metric has made it possible to validate the results obtained.

The purpose of this paper is to demonstrate mathematically the impossibility of achieving a utopian society. Demonstrate that any attempt to correct deviations from a hypothetical trajectory whose ultimate goal is the utopia, increasingly demands more work, including measures that lead to terror, which may even be absolute, leading to the horrible paradox that in seeking paradise hell is constructed.

Scientific tools that the authors have used are: the theory of the system linkage, alysidal algebra, kinematic theory and vector analysis.

Myths are the substrate of some complex systems of beliefs and utopia is its ultimate goal. The use of the combination of the theory of trajectories, belonging to the alysidal algebra, the theorem of unintended effects and kinematics theory provides an approximation to deviations suffering utopian ideological currents and their corrections.

This paper is a continuation of other previous papers developing the theory of complex societies.

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 looks at the mathematical and logical development of human society structure.

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 to the system, in as much to the modality faculty its neurocerebral system, because it allows him to make decisions. Transactions of energy, money, merchandise, population, etc. would be the equivalent one to the sanguineous system. These economic transactions and inferential relations, depend, as well, of the existence of a legislative body with their obligations, prohibitions and permissions that regulate them. A Social System Σ may be considered like an alysidal set with an only alysidal element. The authors consider two theories: Enlarged theory and Reduced theory.

This paper is a continuation of two previous papers – Part I published in Kybernetes, Vol. 41 No. 1/2 and Part II published in Vol. 41 No. 5/6 – and develops the theory of deontical impure systems.

This paper aims to refer to a subjective approach to a type of complex system: human ecosystems, referred to as deontical impure systems (DIS) to capture a set of properties fundamental to the distinction between human and natural ecosystems. There are four main phenomenological components: directionality, intensity, connection energy and volume. The paper establishes thermodynamics of deontical systems based on the Law of Zipf and the temperature of information.

Mathematical and logical development of human society structure.

A fundamental question in this approach to DIS is the intensity or forces of a relation. Concepts are introduced as the system volume and propose a system thermodynamic theory. It hints at the possibility of adapting the fractal theory by introducing the fractal dimension of the system.

This paper is a continuation of other previous papers and developing the theory of DIS.

The process of elaboration of the symbolic universe leads to important insights into the role of symbols in understanding human reasoning. Symbols become explanatory axes of universal global realities. Myths were constructed on these explanatory paths forming a superstructure of all belief systems with paraconsistent logic for the symbolism and a symbolic syntax. Myths and symbols are to be found in all cultures. Some of the most powerful and influential ones occur in popular culture since these often have the greatest immediate social impact.

Semiotic and logical development of the symbols is in mythical systems. The dissolution of the myth and the degradation of the myth's symbols constitute a long-drawn-out process in modern Western society and wherever s influence reaches. Myth is a story that may contain symbolic elements, but compared to the symbols or images of the exceptional, myth is characterized by a “story.”

Starting from a minimal definition to define myths and propose the following definition: Myth is a traditional tale that relates memorable and exemplary actions of extraordinary personages in prestigious and distant times, and myths have various forms and functions, perhaps some more clearly defined with a signifier than others, and different approaches can be combined for a better understanding of the myths. Dispensing with such simplistic assertions, and starting from a minimal definition to define myth, myth is a traditional tale that relates memorable and exemplary actions of extraordinary personages in prestigious and distant times.

Any symbol F originates in a unit that has two aspects and functions when the unit is restored. Thus, the symbol is rather “for something” than “of something” and the symbolic objects express the objects' correspondence in one unit or hendiadys. One semantic characteristic of symbols is “recognition”. The symbol F reveals a reality by means of the homogenous association of the signifier and significance in the symbol's constitution; although reality is separate, there is a homogeneous relation between the symbolizing and symbolized in symbolization.

The doctrine of coincidentia oppositorum (the coincidence of opposites), the interpenetration, interdependence and unification of opposites has long been one of the defining characteristics of mystical (as opposed to philosophical) thought. Mystics of various persuasions have generally held that such paradoxes are the best means of expressing within language, truths about a whole that is sundered by the very operation of language itself. Any effort, it is said, to analyze these paradoxes and provide them with logical sense is doomed from the start because logic itself rests upon assumptions, such as the principles of non-contradiction and excluded middle, that are violated by the mystical ideas.

Mathematical development of a dialectical logic with truth-values in a complex field.

The coincidentia oppositorum is a common trope in many religious traditions, particularly those with a mystical or initiatory aspect, and fields of knowledge such as Psychology and Quantum Physics, with wave-particle duality. The aim of this paper is to present a mathematical theory of the coincidence of opposites, and where truth-values are expressed in a complex field. A propositional coincidentia oppositorum algebra is developed.

Although the literature of paraconsistent logics is abundant, the authors think that this is the first time that a dialectical logic has been developed with truth-values belonging to a complex field. The impossibility of reaching an absolute truth from opposite propositions is discovered, both being true, because of the existence of irrational numbers in the truth-values.