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.

ARISTOTLENOTHING CAN BE AND NOT BE AT THE SAME TIME

KANTARISTOTLE MADE THE LOGIC FINISHED

FREGE

CANTOR

RUSSELLTHE 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

(→ ()

(→(V)

(→(→)

→¬

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, A belongs to T ⇒ ¬A doesn’t belong to T

¬A, A belongs to T ⇒ A doesn’t 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

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

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

• 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 OA), for a representation of situations contradictories; and b) in that systems happens the Explosion Principle: (OA OA)OB.