PARACONSISTENT LOGIC AND LEGAL EXPERT SYSTEMS A TOOL FOR by lff30040

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									PARACONSISTENT LOGIC AND LEGAL EXPERT
SYSTEMS: A TOOL FOR JURIDICAL ELETRONIC
              GOVERNMENT
                          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.
          ARISTOTLE
NOTHING CAN BE AND NOT BE AT THE SAME TIME
           KANT
ARISTOTLE MADE THE LOGIC FINISHED
FREGE
CANTOR
             RUSSELL
THE SET OF ALL SETS THAT ARE NOT MEMBERS OF
                 THEMSELVES
• Vasiliev
                                                          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.
JASKOWSKI
                                                         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.
NEWTON C. A. DA COSTA
               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
 “don’t go”. A “classic” robot in presence of
 any contradiction will became trivial, acting
 in a disorder way.
   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
Paraconsistent Propositional Calculus
Paraconsistent Propositional Calculus
Paraconsistent Propositional Calculus
                  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 doesn’t belong to T
    |- A ⇒ A belongs to T
    A, Ao belongs to T ⇒ ¬A doesn’t belong to T
           o
    ¬A, A belongs to T ⇒ A doesn’t 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
            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(Bo) = v(A →B) = v(A->¬B) = 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
                 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
• The representation of rules in conflict, in classical
  systems of deontic logic found two difficulties: a) it
  isn’t 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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