J. Software Engineering & Applications, 2011, 4, 579-584
doi:10.4236/jsea.2011.410067 Published Online October 2011 (http://www.SciRP.org/journal/jsea)
Copyright © 2011 SciRes. JSEA
Paraconsistent Algorithm Extractor of
Contradiction EffectsParaExtrctr
João Inácio Da Silva Filho1, Germano Lambert-Torres2, Luiz Fernando Pompeo Ferrara1,
Maurício Conceição Mário1, Marcos Rosa dos Santos3, Alexandre Shozo Onuki1,
José de Melo Camargo3, Alexandre Rocco1
1Santa Cecilia University, Santos City, Brazil; 2Itajuba Federal University, Itajuba, Brazil; 3Eletropaulo Metropolitano
Eletricidade de Sao Paulo S.A., São Paulo City, Brazil.
Email: inacio@unisanta.br
Received August 27th, 2011; revised September 20th, 2011; accepted October 15th, 2011.
Nowadays networks of analyses based in non-classic logics are used with success in the treatment of uncertainties. The
characteristic of accepting the contradiction in his structure is the main cause of the methodologies based in Paracon-
sistent Logic is ideals for applications in systems of analyses and decision making. In this work we presented an algo-
rithm based in Paraconsistent logic capable to extract in a gradual way the effects of the contradiction in originated
signals of information of uncertain knowledge database. The Algorithm Paraconsistent Extractor of Contradiction ef-
fectsParaExtrctr is formed with base in fundamental concepts of the Paraconsistent Annotated Logic with annotation
of two values (PAL2v) it can be applied in filters of networks of analyses of signal information where uncertain and
contradictory signals can be present. The process of extraction of the effect of the contradiction is always begun by the
largest inconsistency degree among two signals that belong to the group that is in analysis. In the end of the analysis it
is found a consensus value. In this work we presented numeric example and one example of application of the
ParaExtrctr in Load Profile Forecast used in support to decision of the operation in an Electric Power System, but his
application potentiality is demonstrated in several fields of the Artificial Intelligence.
Keywords: Algorithm, Non-Classic Logic, Paraconsistent Annotated Logic
1. Introduction
The contradictions or inconsistencies are common when
we described parts of the real world; however most of the
Expert systems for decision making are based in the bi-
nary classic logic. In spite of the classic logic to be the
base of the whole current technology, it is limited by
rigid binary laws and their algorithms present difficulties
to treat complex systems conveniently where the signals
of information are just constituted of evidences [1-3].
Acting in a faithful way the existent limits in the real
world the signals constitutes evidences arrivals of several
sources, that bring joined several situations ambiguous,
or contradictory that they are unable of be accepted by
the classic logic [4,5]. To solve problems, and to give
appropriate form of treatment front the situations no rec-
ognized by the classic logic, every day has been used new
methods for applications of logics denominated non-
classic, among them, the paraconsistent logics [6,7]. In-
that work we presented an algorithm built from a special
type of logic denominated Paraconsistent Annotated Lo-
gic (PAL) and whose main foundations will be descri-
bed to proceed.
2. The Paraconsistent Logic
Paraconsistent logics (PLs) belong to the class of the non-
classic logics and were idealized by the need of finding
means of giving appropriate treatments to the contradict-
tory situations [8]. In many studies [5,9] and [10] the PLs
presented results that make possible to consider the in-
consistencies in his structure in a no trivial way and for
that, that logic type is shown more favorable in the reso-
lution of problems caused by situations of contradictions
that appear when we worked with the real world. Among
the several types of Paraconsistent Logics exists a class
of Paraconsistente Annotated Logic (PAL) that have an
associated Lattice and were introduced for the first time
in logical programming by Subrahmanian [10-12]. The
Paraconsistent Algorithm Extractor of Contradiction ef-
Paraconsistent Algorithm Extractor of Contradiction Effects—Paraextr
580 ctr
fectsParaExtrctr uses the methods of uncertainty treat-
ment based in an extension of the Paraconsistent Logic
denominated of Paraconsistent Annotated Logic with
annotation of two values (PAL2v) [3,13].
2.1. The Lattice Associated to the Paraconsistent
Annotated Logic
In the Paraconsistent Annotated Logic (PAL) the pro-
positional formulas come accompanied of annotations
[3,10,12]. Each annotation, belonging to a Lattice finite
attributes values to his correspondent propositional for-
mula. To obtain a larger representation a Lattice associ-
ated at LPA is used (Figure 1).
The representation of paraconsistent logical state is
formed by pairs orderly, such that:
An operator is considered ~: |
|, where the operator ~ it
constitutes the “meaning” of the logical negation symbol
of the system that will be analyzed [3,9,11].
If P is a basic formula, then the operator ~: |
|, it is de-
fined as:
,, were,,0,1
 
It is considered then: (
, λ) an Annotation of P.
The orderly pair’s first element represents the Evi-
dence Degree that is favorable to the proposition P, and
the second element represents the Evidence Degree that
is unfavorable or contrary to the same proposition [3].
This way, the intuitive idea of the association of an an-
notation to a proposition P(
, λ) means that the evidence
degree favorable to P is
, while the evidence degree
unfavorable or contrary to P is λ.
Through the methods presented in [3], a system using
the Paraconsistent Annotated Logic receives signals of
information in the form of degrees of evidence with val-
ues that vary from 0 to 1. These values can be placed in
two representing axes of finite lattice (Figure 2(a)) were
calculating the Certainty Degree (DC) and the Contradic-
tion Degree (Dct) through the equations:
 (1)
 (2)
The Contradiction Degree value (Dct) is an indicative
of the inconsistency measure and the certainty degree
value (DC) is considered as the result of the analysis.
3. The Paraconsistent Analysis Nodes—PAN
The element capable of treating a signal that is composed
of one degree of favorable evidence and another of unfa-
vorable evidence (μ1a, μ2a), and provide in its output a
Resulting Evidence Degree, is called basic Paraconsistent
Figure 1. Finite lattice of the PAL2v four states with values.
(a) (b)
Figure 2. Finite lattice of PAL2v and symbol of the para-
consistent analyzer node—PAN.
Analysis Node (PANb). Figure 2(b) shows the represent-
tation of a PANb with two inputs of evidence degree:
μ1 = favorable Evidence Degree of information
source 1.
λ = unfavorable Evidence Degree.
where: λ = 1 – μ2
μ2 is a favorable Evidence Degree of information
source 2.
A lattice description uses the values obtained by the
equation results in the Paraconsistent Analyzer Node
Algorithm [3,13,14] that can be written in a reduced form,
as follows:
1) Enter with the input values.
μ*/ favorable evidence Degree 0 μ 1
λ*/ unfavorable evidence Degree 0 λ 1
2) Calculate the Contradiction Degree.
3) Calculate the Certainty Degree.
4) Calculate the distance d of the Paraconsistent logical
state into Lattice.
1D Dd 
5) Compute the output signal.
If d 1 Then do S1= 0.5 and
= φ: Indefinite logical state
go to the steep 10
Or else go to the next step
Copyright © 2011 SciRes. JSEA
Paraconsistent Algorithm Extractor of Contradiction Effects—Paraextr 581
6) Calculate the real Certainty Degree.
If DC > 0 D
CR = (1 – d )
If DC < 0 D
CR = (d – 1)
7) Present the output.
Do S1 = D
8) Calculate the real Evidence Degree.
9) Present the output.
Do S1 =
μER and S2 = Dct
10) End.
The Systems with the Paraconsistent Analysis Nodes
(PAN) deal with the received signals through algorithms,
and present the signals with a Certainty Degree value and
a Contradiction Degree value in the output [3].
4. The Paraconsistent Algorithm Extractor
of Contradiction EffectsParaExtrctr
The Paraconsistent Algorithm Extractor of Contradiction
effects (ParaExtrctr) is composed by connections among
PANs. This configuration forms a Paraconsistent Analyze
Network capable to extract the effects of the contradict-
tion in gradual way of the signals of information that
come from Uncertain Knowledge Database. The hypo-
thesis of extraction of the effects of the contradiction has
as principle that; if the first treated signals are the most
contradictory, then the result of the paraconsistent analy-
sis will converge for a consensual value. In his typical
operation the ParaExtrctr receives a group of signals of
information represented by Degrees of Evidence (μE) the
regarding certain proposition P and, independently of other
external information, it makes paraconsistent analysis in
their values where, gradually, it is going extracting the
effects from the contradiction to remain as output a single
resulting Real Evidence Degree μER.
The μER is the representative value of the group of in-
put signals after the process of extraction of the effects of
the contradiction. The Figure 3 shows the representation
of the algorithm Extractor of Contradiction effects that
uses a network of three PANs.
The description of the ParaExtrctr Algorithm is shown
to proceed.
1) Present n values of Evidence Degrees that it com-
poses the group in study.
μ = (μA, μB, μC, ···, μn ) */Evidence Degrees 0 μ
2) Select the largest value among the Evidence De-
grees of the group in study.
μmaxA = max (μA, μB, μC, ···, μn)
3) Consider the largest value among the Evidence De-
grees of the group in study in favorable Evidence Degree.
μmaxA = μsel
Figure 3. Paraconsistent algorithm extractor of contradic-
tion effects (ParaExtrctr).
4) Consider the smallest value among the Evidence
Degrees of the group in study in favorable Evidence De-
μminA = min (μA, μB, μC, ···, μn)
5) Transform the smallest value among the Evidence
Degrees of the group in study in unfavorable Evidence
1 –
μminA = λsel
6) Make the Paraconsistent analysis among the se-
lected values:
μR1 = μsel λsel */where is a paraconsistent
action of the PAN */
7) Increase the obtained value μR1 in the group in study,
excluding of this the two values μmax and μmin, selected
Gμ = (μA, μB, μC, ···, μn, μR1) – (μmaxA, μminA)
8) Return to the item 2 until that the Group in study
has only 1 element resulting from the analyses.
Go to item 2 until Gμ = (μER)
5. Application forms of the ParaExtrctr
For the paraconsistent analysis that it uses the ParaExtrctr
Algorithm the information signals in the form of meas-
ures of physical greatness are obtained in uncertain data-
base. That information is representatives of attributes
gotten usually through subjective answers that just gener-
ate evidences regarding the analyzed proposition P. In
that way, the obtained information can come represented
by resulting numbers of quantitative analyses exposed in
tables and in the percentile form. To receive the treatment
through the ParaExtr ctr Algorithm the values of the Da-
tabase are normalized inside of an Interval of Interest—or
Discourse Universe—resulting in Evidences Degrees
valued between 0 and 1. At the end of the analysis made
Copyright © 2011 SciRes. JSEA
Paraconsistent Algorithm Extractor of Contradiction Effects—Paraextrctr
Copyright © 2011 SciRes. JSEA
5.2. The ParaExtrctr Algorithm Applied in
Forecast of Load Profile of Electric
Power Systems
by the ParaExtrctr Algorithm the Resulting Evidence De-
gree will be the real representative of the group com-
posed by n Degrees of Evidence applied in the inputs.
With the value of the Resulting Evidence Degree a proc-
ess of undo normalization can be made to recover it re-
sulting as measure of the analyzed greatness. The Figure
4 presents the flow of signals in the paraconsistent analy-
sis with ParaExtrctr Algorithm.
The innovative form of doing treatment of signals allows
Table 1. Measured different values in four monitored points.
Monitored points A B C D
Measurement instants t1 t2 t3 t4
7.800 7.900 6.800 8.300
7.956 8.900 7.200 7.300
6.960 8.100 7.100 9.300
Measured values
7.750 7.180 8.100 8.700
5.1. Explanatory Example of Use of the
ParaExtrctr Algorithm
We presented to follow an example of application of the
ParaExtrctr Algorithm treating contradictory values ob-
tained in a Database of an Electric Power System [15,16].
Consider that a Database presents values of the greatness
electric of potency in the form of measurements obtained
in quilowatts (Kw). An example of different measure-
ments in the monitored points is shown in the Table 1.
Table 2. Evidence Degrees in four monitored points and the
Resultant Evidence Degree after extract the effects of con-
The monitored points are obtained of different type of
measuring devices. These measurement devices should
bring measures of same values and, in some cases, dif-
ferent values. This happen by several reasons, as differ-
ences of quality of the measure devices, flaws in the ob-
taining of the values and mistakes of readings. In this
process, in certain instant can happen that contradictory
values exist in the t1 time, or in t2, or in t3 and in t4.
Monitored points A B C D
Measurement instants t1 t2 t3 t4
0.7000 0.7250 0.45000.8250
0.7390 0.9750 0.550005750
0.4900 0.7750 0.52501.0000
Evidence Degrees
Universe Discourse
5.0 Kw 9.0 Kw
Lineal variation
0.6895 0.5450 0.77500.9250
ParaExtrctr Algorithm action
μER1 μER2 μER3 μER4
Resultant Evidence
Degrees to each instant t0.6648 0.7235 0.55080.8018
Resultant Evidence
Degrees in four instants:
t1, t2, t3 and t4
The definition of the Universe of Discourse and the
variation of the greatness in the interval will be important
for the generation of the Evidence Degrees.
For that explanatory example we will consider a Uni-
verse of Discourse of 5.0 up to 9.0 Kw with linear varia-
tion. The Table 2 shows the result of the application of
the ParaExtrctr Algorithm in the extraction of the contra-
diction effects in the four signals in each one of the time t
and in four times t1, t2, t3 and t4.
Figure 4. Signals flow in an application of the paraconsistent algorithm extractor of contradiction effects (ParaExtrctr).
Paraconsistent Algorithm Extractor of Contradiction Effects—Paraextr 583
that the ParaExtrctr Algorithm can be applied in several
fields where it is necessary the use of Artificial Intelli-
gence techniques. As a real application in AI systems it is
mentioned as example, a forecast accomplished in the
Substation ETD Pattern 40/60 MVA of the Power Elec-
tric System of concessionary of Electric power that acts
in Brazil [2,15].
In this study, as source of information for the forecast,
is used a historic database where are considered the val-
ues of load that are related to the secondary windings of
the two input transformers.
The four signals, that bring the values regarding to the
three phases (RST), receive the paraconsistent logical
treatment by the ParaExtrctr Algorithm in the beginning
Once obtained the values of the load, these are nor-
malized and they will be inside of the closed interval [0,1]
in the set of the real numbers. In that way, the data will
be ready for the treatment according to the PAL2v foun-
The extraction of the contradiction effect in three
phases (RST), are shown in the graph on the Figure 5.
The normalized values of the three phases are applied
in the module of the ParaExtrctr Algorithm and, after the
paraconsistent treatment logical results in an only value
of Evidence Degree of the load profile. In this way the
graphic scenery of the electric load Profile is created ac-
cording to shown in Figure 6.
Several tests were made comparing the values of the
historic database with the values obtained through the
ParaExtrctr Algorithm, showing a great approach of the
real results. Now this Load Profile Forecast System is
Figure 5. Values graphs of three signals that received treatment of the ParaExtrctr algorithm and they will serve as reference
for the elaboration of forecast of profiles loads in the Power Electric System.
Figure 6. Result graph of the load profile forecast using the ParaExtrctr algorithm where the curve is made with values of re-
ulting evidence degrees. s
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Paraconsistent Algorithm Extractor of Contradiction Effects—Paraextr
584 ctr
being used for training and support to the operation of
Power Electric System in Eletropaulo CompanyBrazil.
6. Conclusions
In this work we presented the ParaExtrctr Algorithm that
is capable to extract contradiction effects in groups of evi-
dence signals the regarding certain Proposition through
basic concepts of the Paraconsistent Logic. In the end of
the analysis, the ParaExtrctr Algorithm presents as result
a Real Evidence Degree representative of the Evidence
Degree group. In that process of Paraconsistent analysis
the ParaExtrctr Algorithm uses the denominated PAN-
Paraconsistent Analysis Nodes. In a gradual way the
PANs filter the effects of the contradiction in the signals
of information until that it is found the Degree of Evi-
dence resulting from the group. The results demonstrate
that the ParaExtrctr algorithm has capacity to remove of
database the values tuneless or contradictory.
The extraction process reduces the effects of the in-
consistencies and it presents as answer a closer represen-
tative value of the reality. In that way, with the Para-
Extrctr Algorithm new structures and different configura-
tions of networks of analyses can be formed for treatment
of Uncertainties.
The methods used in this work are based in a special
Paraconsistent logic denominated of PAL2v and they
have been applied with success in the determination of
patterns, and in making decision systems, as well as in
different areas of the knowledge where are necessary the
performance of Intelligent Systems.
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