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

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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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ARISTOTLENOTHING CAN BE AND NOT BE AT THE SAME
TIME
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KANTARISTOTLE MADE THE LOGIC FINISHED
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FREGE
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CANTOR
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RUSSELLTHE 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
• (?o L bo) ? (? L b)o
• (?o L bo) ? (? V b)o
• (?o L bo) ? (? ? b)o
• ?o ? (?)o

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Paraconsistent Propositional Calculus
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Paraconsistent Propositional Calculus
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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, Ao belongs to T ? A doesnt belong to T
• A, Ao belongs to T ? A doesnt belong to T
• A ? B belongs to T ? B belongs to T
• Ao, Bo belongs to T ? (A ?B)o, (A L B)o, (A V
  B)o belongs to T

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Validation Function

• A validation of C1 is one function v F -gt
  0,1, as A and B are any formulas
• v(A) 0 ? v(A) 1
• v( A) 1 ? v(A) 1
• v(Bo) v(A ?B) v(A-gtB) 1 ? v(A) 0
• v(A ?B) 1 ? v(A) 0 ou v(B) 1
• v(A L B) 1 ? v(A) v(B) 1
• v(A V B) 1 ? v(A) 1 ou v(B) 1
• v(Ao) v(Bo) 1 ? v((A ?B)o) v((A L B)o)
  v((A V B)o) 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(Ao) 0 ? v(A) v(A) 1
• v(A) 0 ? v(A) 0 e v(A) 1
• v(Ao) 1 ? v((A)o) 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.