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PARACONSISTENT LOGIC AND LEGAL EXPERT SYSTEMS: A TOOL FOR JURIDICAL ELETRONIC GOVERNMENT

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Origin During many centuries the logic of Aristotle (384-322 a.C.) served as foundation for all the studies of the logic. Between 1910 and 1913, the Pole Jean Lukasiewicz (1876-1956) and the Russian Nicolai Vasiliev (1880-1940) had tried to refute the Principle of the Contradiction.

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ARISTOTLE NOTHING CAN BE AND NOT BE AT THE SAME TIME

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KANT ARISTOTLE MADE THE LOGIC FINISHED

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FREGE

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CANTOR

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RUSSELL THE SET OF ALL SETS THAT ARE NOT MEMBERS OF THEMSELVES

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Vasiliev

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Origin S. Jaskowski (1906-1965), a disciple of Lukasiewicz, presented in1948 a logical system that inconsistency could be applied. The system of Jaskowski had been limited in part of the logic, that technical is called propositional calculation, not having perceived the possibility of the paraconsistents logics in ample direction, or either, applied to the calculation of predicates.

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JASKOWSKI

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Origin Independently of Jaskowski (whose works had been publish in pole) and motivate by matter of philosophy and maths, the Brasilian Newton C. A. da Costa (1929-), at that time professor of UFPR, started in 1950 studies of a logical system that could accept contradictions. The systems of da Costa (the systems C) are more extensive that the systems of Jaskowski.

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NEWTON C. A. DA COSTA

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Application Expert systems: in medicine, when two or more diagnostics have contradictions made by different doctors. Robotic: the robot can be program with a lot of different sensors, and these sensors could create informations with contradictions: a optical visor may not detect a wall of glass, saying free to go while other sensor could detect it, saying dont go. A classic robot in presence of any contradiction will became trivial, acting in a disorder way.

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Paraconsistent Propositional Calculus In the beginning, the same of the classical logic ( ( ) ( ( V ) ( ( ) ¬

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Paraconsistent Propositional Calculus

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Theorem 1 If T is not trivial maximal and A and B are formulas : T |- A A belongs to T A belongs to T ¬ * A doesnt belong to T |- A A belongs to T A, A belongs to T ¬ A doesnt belong to T ¬ A, A belongs to T A doesnt belong to T A B belongs to T B belongs to T A, B belongs to T (A B), (A B), (A V B) belongs to T

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Validation Function A validation of C1 is one function v: F -> {0,1}, as A and B are any formulas: v(A) = 0 v( ¬ A) = 1 v( ¬ ¬ A) = 1 v(A) = 1 v(B ) = v(A B) = v(A-> ¬ B) = 1 v(A) = 0 v(A B) = 1 v(A) = 0 ou v(B) = 1 v(A B) = 1 v(A) = v(B) = 1 v(A V B) = 1 v(A) = 1 ou v(B) = 1 v(A ) = v(B ) = 1 v((A B) ) = v((A B) ) = v((A V B) ) = 1

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Theorem 2 If v is a validation of C1, v has the following property: v(A) = 1 v( ¬ * A) = 0 v(A) = 0 v( ¬ * A) = 1 v(A ) = 0 v(A) = v( ¬ A) = 1 v(A) = 0 v(A) = 0 e v(~A) = 1 v(A ) = 1 v(( ¬ A) ) = 1 v(A) = 1 v(A) = 1 ou v( ¬ A) = 0

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The representation of rules in conflict, in classical systems of deontic logic found two difficulties: a) it isnt possible in that system expressions like (OA O A), for a representation of situations contradictories; and b) in that systems happens the Explosion Principle: (OA O A) OB.

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