# 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
ARISTOTLE
NOTHING CAN BE AND NOT BE AT THE SAME TIME
KANT
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.
Origin

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
NEWTON C. A. DA COSTA
Application
Expert systems: in medicine, when two or
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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