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TRANSDISCIPLINARITY AND LOGICS Itala M. Loffredo D’Ottaviano The Group of Applied and Theoretical Logic Centre for Logic, Epistemology and the History of Science Philosophy Department State University of Campinas II WORLD CONGRESS ON TRANSDISCIPLINARITY VILA VELHA, BRAZIL SEPTEMBER

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LOGIC Logic, the science of the deductive reasoning, studies the consequence relation, treating of the valid inferences, that is, the inferences whose conclusions have to be true when the premisses are true ones. Therefore, logic can be considered as “the study of the reason”, “the study of the reasoning”.

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So, the objective of logic consists of the mention and the study of the principles used in the deductive reasoning. Under this conception, logic is called deductive logic.

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Meanwhile, we can consider another logic, the inductive logic, that does not treat of the valid inferences, but treats of the probable inferences.

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Let us consider the following arguments: The sun has been born every day. Then, the sun will be born tomorrow. 80% of the interviewed people will vote in the candidate X. Then, 80% of all the electorate will vote in X. This vaccine well worked in monkeys. This vaccine well worked in pigs. Then, this vaccine will well work in human beings.

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Those arguments are not deductive. Even if the premisses are true, they do not logically imply the conclusion. Meanwhile, if the premisses are true, then the conclusion is plausible, that is, it is probably true.

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Peirce Abductive Reasoning

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Contemporary, logic has been transformed into a mathematical discipline, the mathematical logic, with special characteristics – deductive, it is the study of the type of reasoning done by the mathematicians.

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The contemporary logicians build appropriate artificial languages to deal with the consequence relation. Such languages have two relevant dimensions: the sintactical dimension and the semantical dimension.

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In order to work in a formal theory, it is necessary to explicitate its language. That is, its symbols and the combination rules to which they are submitted for the construction of the terms and the formulae of the language. The axioms and the deduction rules of the theory are specified among the well formed formulae of the language. From the axioms and by using the deduction rules, the theorems of the theory are proved.

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The combinatoric dimension of a language is called syntactic dimension. The semantic dimension of a language takes into consideration the extra- linguistic objects to which the symbols and expressions of the language refer to, and their meaning. It deals with the concepts of structure, validity of formulae and model.

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So, the contemporary theories, constructed over languages, axioms and deduction rules, are constituted by the formal theory – its axiomatics -, and by its semantics. The results relative to the completeness, consistency and decidability of a theory are important meta-theorems that establish the relations between these two dimensions.

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Meanwhile, until the beginning of the 20th Century, there was a unique logic, pure, formal or theoretical, founded by Aristotle (384 to 322 b.C.), and whose most important systematizer was Frege (1848 to 1925).

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Most of Aristotle’s relevant contribuition to logic is in the group of works known by Organon, especifically in the Analytica Priora and in the De Interpretatione.

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Aristotle created the theory of silogisms and axiomatized it. He also initiated the development of the modal logic.

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The theory of silogisms constitutes the first formal system of the history of logic. Contemporarily, we can interpret it as a fragment of the first-order predicate calculus.

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The modern logic initiated in the 17th Century, with Leibniz. Leibniz’ programme looked at the construction of a universal language, based on an alfhabet of thougt. This language would propitiate a fundamental knowledge of everything.

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Leibniz added to his work the project of the construction of calculus ratiocinatur, that is, a calculus of the reason. In spite of Leibniz’ programme being not theoretically executable, the calculus ratiocinatur constitutes an important forerunner of the methodology of contemporary logic.

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Menwhile, most of Leibiniz’ contributions to logic remained not published during his life and unknown until the beginning of the 20th Century.

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Immanuel Kant contributed very little to logic. But, in the Preface of his Kritik der reinen Vernunft, edition of 1787, he explicitly declares that the logic had not given any important step, neither ahead nor behind, since Aristotle – so, it seemed to be complete and finished.

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The true founder of the modern logic was Gottlöb Frege.

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In 1879, the essential steps for the introduction of the logistic method were given by Frege in his Begriffsschrift. This book contains, for the first time, the propositional calculus in its modern logistic form.

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Frege introduces, for the first time, the distinction between language and meta- language. In 1884, Frege addopts the thesis – logicism – that arithmetics is a branch of logic, in the sense that every term of arithmetics can be defined from the logical terms and that every theorem of arithmetics can be proved from the logical axioms.

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In 1874, George Cantor creates the set theory. He publishes his first work on a new theory of the infinite, where a collection of objects, even infinite, is conceived as a complete entity.

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For Cantor, a set was intuitivelly a collection of elements that satisfy a given property. At the beginning of the 20th Century, this apparently naïve acceptance of any collection as a set, propitiated the appearance of paradoxes in the foundations of the nascent set theory.

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Type Theory and Set Theories In 1908, in the opening section of the “IV International Congress of Mathematics”, hold in Rome, Poincaré claimed the mathematical community to find a solution to the paradoxes crisis, that seemed to shake the foundations of mathematics.

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Zermelo and Russell already worked looking for the fundamental principles that could underlie a consistent theory, without contradictions.

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Russell and Whitehead publish their Principia Mathematica in 1910, 1912 and They introduce the ramified type theory, a system that establishes a hierarquy of types and collections. The type theory presents a general solution for the problem of the paradoxes.

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The set theory, nascent at the beginning of the 20th Century, had a sufficient basis to resist to the crisis of the paradoxes. The set theories present a partial solution to the problem of the paradoxes, by eliminating the syntactic paradoxes of mathematics. They constitute strong systems for the foundation of mathematics.

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The Classical Logic The classical logic, in its elementary part, essentially deals with the logical connectives of negation, conjunction, disjunction, implication and equivalence; it deals with the existencial and universal quantifiers and with the equality predicate; and deals with some of their extensions. It is characterized as a logic of propositions.

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From Frege’s work, the classical logic got an extense, consistent and almost definitive form in the Principia Mathematica of Whitehead and Russell. In its contemporary status it is powerful and contains all the old Aristotelian silogistic, conveniently reformulated. Besides, in a certain sense, the traditional mathematics can be reduced to the classical logic, for it can be definible from the idea of set.

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In its contemporary clothes, logic is considered as a deductive formal system, built over a formal language, that would have the charge of eliminating interpretative doubts.

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The Classical Propositional Calculus - CPC The basic elements of the classical propositional calculus are the propositions, that are expressions that admit a truth value: (0) false or (1) true.

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From the simple or atomic propositions we can form the compound or molecular propositions, by using the linguistic connectives. The expressions ‘not’, ‘and’, ‘or’, ‘if...then’, and ‘if and only if’ are important connectives.

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Every propositional variable is a formula, called an atomic formula. From these atoms, if A and B are atomic formulas, then the following expressions are formulas: A (the negation of A) A B (the conjunction of A and B) A B (the disjunction of A and B) A B (the conditional from A to B) A B (the biconditional between A and B).

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The most intuitive and simple way to ascribe a semantics to the classical propositional calculus is through the valuation functions: v: {p 1, p 2, p 3,...} {0, 1}

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The Formal System CPC is determined by: Axioms and Deduction Rule (or Inference Rule)

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The Basic Laws of the Aristotelian Thought 1.Principle of (Non-) Contradiction 2. Principle of the Excluded Middle 3. Reflexivity of the Identity

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Inference Rule The only inference rule is the Rule of Modus Ponens (MP): If A and B are formulae then, from A and A B we obtain B.

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Classical Logic and the Arising of Non- Classical Logics The mathematics of the 19th Century – one of the golden periods of mathematics – strongly influenced the culture and the thought of the 20th Century. It either directly or indirectly contributed for the arising of the mathematical logic and, especially, for the arising of non-classical logics. One of the fundamental marks was the appearance of the non-Euclidean geometries.

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The usual set theory (and Russell’s type theory), over which arithmetics can be founded – and so all the traditional mathematics -, maintains the classical logic with its basic principles – the basic laws of the Aristotelian thought – as its underlying logic.

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Meanwhile, the paradoxes of the set theory and some non-solved questions relative to the concept of the infinite still left to the logicians several problems concerning the foudations of mathematics.

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Already at the end of the 19th Century some pioneer works, looking for non- Aristotelian solutions for some logical questions, were forerunners of the non- classical logics in general. At the first decades of the 20th Century, several philosophers and mathematicians, motivated by distinct questions and objectives, created new logical systems, different of the Aristotelian logic.

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We can assert that the non-classical logics differ of the classical logic by: i) They can be based on languages richer in manners of expression; ii) They can be based on completely distinct principles; or iii) They can have a distinct semantics.

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We can consider two main categories of non-classical logics: the complementary logics to the classical logic, and the alternative logics to the classical logic, or heterodox logics.

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The complementary logics do not infringe the basic principles of the classical logic, but they only enlarge and complement its scope. Examples of complementary logics: Modal Logics Deontic Logics Epistemic Logics Temporal Logics

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The heterodox logics, rivals of the classical logic, were conceived as new logics, determined to substitute the classical logic in some domains of the knowledge. They raze basic principles of the classical logic.

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Non-reflexive Logics: they are the heterodox logics in which the Law of the Reflexivity of the Identity does not work. Example: Quantum Logic Quantum Computation

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Paracomplete Logics: they are the logics in which the Principle of the Excluded Middle is not valid. Examples: Intuitionistic Logics Many-valued Logics

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Paraconsistent Logics: the Principle of (Non-) Contradiction may be not valid, in general.

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A theory is said to be consistent if there is not any formula A of its language such that A and the negation of A are theorems; otherwise, the theory is inconsistent. A theory is said to be trivial if every formula of its language is a theorem. Every deductive theory based on the classical logic is inconsistent if, and only if, it is trivial.

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A logic is said to be paraconsistent, if it can be used as the underlying logic to inconsistent but non-trivial theories, that are called paraconsistent theories.

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In paraconsistent logics, the scope of the Principle of (Non-) Contradiction, is in a certain, restricted.

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In fact, in paraconsistent logics, the Principle of (Non-) Contradiction, in the form ¬(A ⋀ ¬A) is not necessarily non-valid, but, in every paraconsistent logic, from a formula A and its negation ¬A it is not possible, in general, to deduce any formula B.

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There exist several other types of non- classical logics, as for example: Relevant Logics Linear Logic Non-Monotonic Logics Fuzzy Logics (Zadeh, 1976

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Modern logic has very much evolved. And, relatively to certain types of logic, it would be difficult to identify them either as complementaries of the classical logic or heterodox logics..

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The creation of the non-classical logics originated important philosophical problems. But it seems that their meaning has not yet been deeply debated.

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Paraconsistent Logic The Brazilian Newton Carneiro Affonso da Costa is internationally considered the founder of paraconsistent logic.

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In the years 1950, without knowing Jaśkowski’s work, da Costa began to develop his ideas about the importance of the study of the contradictory theories.

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In 1958, da Costa proposes the following Tolerance Principle in Mathematics From the syntactical and semantical point of view, every theory is acceptable, since it is not trivial.

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In 1963, da Costa’s ideas were completely developed, when he initiated the publication of a series of papers containing his hierarquies of logics for the study inconsistent but non-trivial theories.

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SELF-ORGANIZATION From some decades, new developments in areas such as logic, the theory of information, cybernetics, physics, chemistry, cellular and molecular biology, ecology and cognitive science have raised a retaken of philosophical and scientific reflection about the notions of order, disorder, organization, disorganization and complexity, allowing us to conjecture the overcome of classical theoretical oppositions.

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In 1986, at the Centre for Logic, Epistemology and the History of Science of UNICAMP, a transdisciplinary group of researchers, under the coordenation of the philosopher Michel Debrun, began to study problems related to the notions of order, disorder, crisis, caos, information, self- organization, hetero-organization, autopoiesis, self-reference, complexity, systemics.

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One of the aims of the work of the group was to propose a definition for the term self-organization, in order to capture the meaning of the phenomena usually identified as self-organized ones.

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Is there anything as self-organization?

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An organization or a form is self-organized, when it produces itself. The new, the emergent, must have its origins at the level of the own process; neither only in the start conditions, nor in the inter-change with the surroundings.

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Debrun’s Definition There is self-organization every time when, from a meeting among actually – and not simply analitically – distinct elements, it is developed an interaction without supervisor – or without omnipotent supervisor – such an interaction that eventually leads either to the constitution of a “form” (organization), or to the re- structuration, by “complexification”, of a already existent form (organization).

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SYSTEMICS and SELF-ORGANIZATION In a recent joint paper with Ettore Bresciani, we began the study of the phenomena of organization and self-organization, based on fundamental notions, concepts and definitions that make part of the theory of systems, the systemics.

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In order to treat a problem from the perspective of a systemic approach, some principles and conditions must be recognized:

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1. The existence of the system with an underlying structure, constituted by a non- empty set (its universe) of elements (its individuals) and by the relations between these elements; and with a functionality.

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2. The characterization of the internal, external and frontier elements.

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3. The existence of the properties of sinergy, globality and novelty.

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4. The presence of a subject – observer of the system -, of a complex nature and who may be an internal, external or frontier element of the structure of the system.

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5.The possibility of the system to receive from the exterior (the surroundings), to internally transform and to transmit to the exterior, through a frontier, energy, material and information.

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6.The identification of relations, between the elements of the system, of distinct degrees of complexity.

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7.The identification of a property of the system, characterized by its structure and by its working, denominated organization, that leads the behavior of the system.

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8. The existence of the teleological properties and of the equifinality of the system.

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9.The creation of perturbation and restrictive conditions, of determined and undetermined characteristics, in the system, due to the interaction with the surroundings through its frontier.

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10.The necessity of the existence of a field of influence (or of forces) in order to provoque a flow of activities.

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11.The possibility of maintaining the structural and functional equilibrium, that is, the maintenance of the state of the system in its relations with the surroudings, through the mechanism of adjustment.

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12.The possibility of change of state, with the emergence of a new state, that characterizes either the creation or the evolution, through the mechanism of structural and functional adjustment.

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13.The possibility of the presence of the phenomenon of self-organization, as a consequence of the interaction between the pre-determined activities of the system, and the spontaneous and autonomous activities of the elements of the system, in a circular process.

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14. The possibility of transformations, through creative processes, that may be consequences of the phenomenon of self- organization.

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LOGIC AGAIN With the crisis of the paradoxes at the beginning of the 20th Century, the publication of the Principia and the creation of the set theory, Kant’s “finished science” had meaningful transformations, that motivated a big development, with the creation of several research areas, and under certain circunstances characterized logic as a discipline of mathematics.

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The development of the non-classical logics in general, has opened several research areas and has propiciated the solution of important questions of the foundations of science.

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Several applications of many-valued logics have been studied and developed, such as to the theory of electric circuits, to linguistics, to computers programming and to the theory of probability.

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Nowadays, the many-valued logics have been innovatingly applied to the treatment of the information in conditions of uncertainty and to the problems there originated, including those concerning to computability and complexity.

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Relatively to modal logic and its development, dynamic logic deserves to be mentioned, that is, the logic that represents processes of computation.

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The paraconsistent logics, looked as formal theories that support inconsistent but non- trivial theories, constitute a natural solution for the treatment of the question of the tolerance to fails: an intelligent system has to work under imprecision of language, of all kinds of especifications, and inclusive under imprecisions of consistency.

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Besides the new and not foreseen uses of the classical logic, it is not difficult to perceive the connexion between the non-classical logics and the artificial intelligence, that has among its interests the reasoning processes that can be formulated and controlled in the computable mathematical universe and, so, must be naturally based on logic.

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We are living a new revolution of logic, with the emergence of new paradigms – connected to creation of the computers and of new areas of science.

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A relevant question then arised: The rigorous study of the human intelligence and of the cognitive processes, and the formalization of creativity and of the process of human decision.

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In several of his papers, da Costa argues that the human reason constitutes itself through history, especially and mainly following the contingency originated by the scientific progress.

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In this sense, the a priori nature of reason seems to be relative. Reason becomes a constitutive element of the culture of a given epoch, so, it has cultural and social connotations, relatively to its own history. In spite of the ambiguity of the term, da Costa considers that reason is dialectical, evolving according to the advance of science.

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The reason can not be codified a priori, via a determined fixed logical system.

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We are not derogating the classical Aristotelian logic, contrarily, we of course know the enormous gama of situations whose analysis explicitly depends on it.

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But, from the arising of the non- classical logics, and with the new paradigm that they conjecture for the 21th Century, we know that there not exists “one” logic, but a better and more adequate logic for every type of problem.

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CLAIM NOWADAYS, LOGIC IS THE COMMON DENOMINATOR OF TRANSDISCIPLINARITY

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“Nowadays, logic is one of the most exciting branchs of knowledge... and one of the greatest cultural revolutions of of our epoch was the construction of the non-classical logics, particularly of the heterodox logics, such a revolution similar to that provoqued by the discovering of the non-Euclidean geometries, in the 19th Century.” (da Costa)

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Or even, parodying Shakespeare: “There is, between the heaven and the earth, more logics than your vain philosophy dreams!”

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