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A type of Inconsistent Mathematics structured on Paraconsistent Logic (PL) and that has, as the main purpose, the study of common mathematical objects such as sets, numbers and functions, where some contradictions are allowed, is called Paraconsistent Mathematics. The PL is a non-Classical logic and its main property is to present tolerance for contradiction in its fundamentals without the invalidation of the conclusions. In this paper (part 1), we use the PL in its annotated form, denominated Paraconsistent Annotated Logic with annotation of two values—PAL2v for present a first-order Paraconsistent Derivative. The PAL2v has, in its representation, an associated lattice FOUR based on Hasse Diagram. This PAL2v-Lattice allows development of a Para-consistent Differential Calculus based on fundamentals and equations obtained by geometric interpretations. In this first article it is presented some examples applying derivatives of first-order with the concepts of Paraconsistent Mathematics. In the second part of this work we will show the Paraconsistent Derivative of second-order with application examples.

There is no doubt that one of the greatest achievements in the area of applied mathematics was the Differential Calculus. The construction of Differential Calculus demonstrates the genius of Newton and Leibniz, as well as other physicists, philosophers and mathematicians who preceded or succeeded them. However, in Infinitesimal Calculus, when using the infinitesimal ε we know that even being infinitely small it is not zero, but regardless of this fact that ε simply disappears at the end of calculation [

The Classical Logic, which emerged in Greece with the ancient philosophers, brings in its fundamentals bi- nary strict laws that do not consider the existence of contradiction, and therefore do not respond well to certain conditions frequently encountered in the real world. From this, it appears that there is the need of finding new ways to design models of physical systems which may prove more efficiency to respond to analysis, particularly those dealing with boundary extreme conditions that are based on information signals which can be contradic- tory [

In this paper, we use the PAL in its structural form in which an annotation of two values is used to perform a type of Differential Paraconsistent Calculus applied in solving problem related to physical systems. The PAL treating information signals in its special form, called Paraconsistent Logic with annotation of two values (PAL2v), allows extracting from the Newton’s quotient, found in the deduction of the differential calculus. All the information is necessary and sufficient to effect the derivative of first- and second-order and apply them to physical systems with good results without ignoring the action of the infinitesimal [

Into the family of the non-Classical logics we have Paraconsistent logics whose main feature is the revocation of the principle of non-contradiction. In formal way [

Therefore, Paraconsistent Logic is a non-Classical Logic repealing the principle of non-contradiction and al- lows the processing of conflicting information in its theoretical structure. Among the precursors of Paraconsis- tent Logic is the polish logician J. Lukasiewicz and the Russian philosopher N.A. Vasilev that independently suggested the possibility of a logic that restrict, for example, the principle of contradiction. The initial systems of Paraconsistent Logic containing all logical levels, involving propositional calculations, of predicate and de- scriptions, as well as logic from superior order is due to N.C.A. Costa (1954 onwards) [

Paraconsistent Annotated Logic―PAL, also known as evidential logic ετ, belongs to the family of Paraconsis- tent Logics and can be represented in a particular way, through a four-vertices Lattice (Lattice FOUR). In intuitive mode the constants of annotation represented in its vertices will give connotations of Logical states related at propositions P [

An atomic proposition of PAL-logic language can be represented by

In [^{2}, where [0,1] indicates the closed unitary real interval. For this set is applied one relation of order, so defined:

Properties:

1) "

2) "

3) "

4) "

5) "

6) "

As proposed in [

Geometric Modeling provides a mathematical description of a geometric object―point, line, conic section, sur- face, or a solid. Geometric transformation is very important for computer graphics, enabling us to manipulate the shape, size, and location of the object [^{2}.

Consider that Unitary Square on Cartesian Plane (USCP) (

degree of evidence unfavorable (λ) in the y-axis. In this system certain annotation (m, λ) can be identified with the point of the plane in another k system with representation in Hasse Diagram (

The Paraconsistent geometric transformations are made between the USCP-Lattice k (

1) Expansion

Expansion of the scale consist change of scale enables the change object size.

Consider a point P(x, y) and P'(X, P) the coordinates of the point after scaling [

Is made an increase of the scale at USCP-Lattice k seen in the

Therefore, being:

2) Rotation

The mathematical expression of the rotation of an object from the origin a certain angle θ is considered from a point P(xp, yp), such that:

For Paraconsistent transformation is made in the USCP-Lattice k a Rotation of 45˚ relating to the origin, therefore:

From

3) Translation

Consider a geometric object represented by a set of points Pi belonging to R^{2} [

adding whole amounts to their coordinates. So, be a point P(x, y) which will be carried out an operation of translation and is P' the coordinates of the point after the translation one can define the function T as being:

With:

Therefore:_{3} obtained in the ordered pair of the transformation in Equation (1) is called Certainty Degree (D_{C}). Therefore, the Certainty Degree is achieved by:

where: m Favorable Evidence Degree.

λ Unfavorable Evidence Degree.

Its values, which belong to the set Â vary in closed interval −1 to +1 and are in the horizontal axis of the PAL2v-Lattice τ of values called “Axis of degrees of certainty”.

The second term Y_{3} obtained in the ordered pair of the equation of the transformation (1) is called Contradic- tion Degree (D_{ct}). Therefore, the Contradiction Degree is obtained by:

The resulting values of D_{ct} belong to set Â, vary on the closed interval +1 and −1 and are exposed on the ver- tical axis of the PAL2v-Lattice τ called “Axis of contradiction degrees”.

In the PAL2v-Lattice τ when D_{C} results in +1 it means that the Paraconsistent logical State (e_{τ}) resulting from paraconsistent analysis is True, and when D_{C} results in −1 it means that it is False. Similarly, when D_{ct} results in +1 means that the Paraconsistent logical State (e_{τ}) resulting from the paraconsistent analysis is Inconsistent T, and when D_{ct} result in −1 means that it is Undetermined ^. _{τ}) can be correlated to the fundamental concept of state, as studied in physical science and then extended to the model based on Paraconsistent Logic.

Or

where: eτ is the Paraconsistent Logical state.

D_{C} is the Certainty Degree obtained according to the two degrees of Evidence μ and λ.

D_{ct} is the Contradiction Degree found according to the two degrees of Evidence μ and λ.

In a static measurement of observable variables in the physical world, in which is obtained the values of Fa- vorable Evidence Degree (μ) and Unfavorable Evidence Degree (λ) to determine the Certainty Degree (D_{C}) and Contradiction Degree (D_{ct}) is always found a single Paraconsistent Logical State (e_{τ}) related to the two informa- tion signals.

The representation of Observable Variables measured in the physical world and the Paraconsistent Universe represented by PAL2v-Lattice τ can be shown according to the

The Certainty Degree normalized from the Paraconsistent Logical Model is called Resulting Degree of Evi- dence [

Likewise, the normalized Contradiction Degree from the Paraconsistent Logical Model [

The Paraconsistent Logical Model allows the results to be reversible, so once obtained the values of the Certainty Degree (D_{C}) and Contradiction degree (D_{ct}) it is possible to recover the Degrees of Evidence from the equations: From equation (2):

From Equation (3):

From Equation (2):

From Equation (3):

The Newton’s quotient can adapt to a Paraconsistent Logical Model in the form of Paraconsistent mathematical Model will be formed based on the initial concepts of the derivatives [

If a function f is derivable or differentiable [

slope of this line is the derivative of the function f at the point a, and is represented by:

Calculus for the resolutions of problems of Physics the derivative represents the instantaneous variation of a function [

function f at point a, and is represented by

It is considered the _{ }+ h, f(x + h)) is given by the Newton’s quotient:

If the result takes positive values (negative), becoming closer to zero, it means that the sequence of points Qj is approaching the point P on the right (left). When

the derivative of f at the point, and is denoted by:

[

derivative at a point, we say that f is derivable (or differentiable) at this point. Likewise, the equation can be

written as h represents a variation of x, such that:

Therefore, the Newton’s quotient is defined as the incremental ratio of f with respect to the variable x, at the point x [

To establish a method of Paraconsistent Differential Calculus where contradictory values will not be despised it is considered initially the Newton’s quotient Equation (8) that can be written as:

It can be applied to the Newton’s quotient a K factor of normalization that aims to put its values within the limits of the PAL2v-Lattice τ, therefore:

where: K is a normalization Factor, whose action allows the equation to be done as the fundamentals of PAL2v.

With the normalization Factor in equation (10) are identified the degrees of Evidence of PAL2v annotation,

such that:

In Paraconsistent Logical Model the K value must be estimated so that the values of the degrees of Evidence become established within the fundamentals of PAL2v. For this becomes:

It is also possible to calculate by Equation (3) the Contradiction Degree of Newton’s quotient, such that:

In the Paraconsistent Logical Model the value of the K adjustment factor will define Contradiction Degree of the final Newton’s quotient, so its value will indicate where into PAL2v-Lattice τ will be the Paraconsistent Logical State (e_{τ}), defined in the Equation (5).

If in the point considered the maximum value of the function is _{ψ}, then

where:

K_{N} is the Paraconsistent Newton Normalization Factor

The value of the Paraconsistent Derivative of the first-order in the physical world is obtained through reap- plying of Newton Normalization Factor (K_{N}) in the result of Paraconsistent Newton’s quotient:

Thus, it is possible for the Paraconsistent Mathematics to be connected to the equilibrium point, defined by the Paraquantum Factor of quantization (h_{ψ}) of the PAL2v-Lattice τ [

The final value of Paraconsistent Derivative depends on the value of variation in x thus it depends on the in- crement of variable x chosen in the calculation.

It is verified that the location of the Paraconsistent logical State ε was adjusted in the PAL2v-Lattice τ through the Paraconsistent Newton Normalization Factor (K_{N}) and identifies how it is represented any differentiable function

Newton’s quotient, which includes the Paraconsistent Newton Normalization Factor (K_{N}), the analysis will be written as:

or

This normalization allows for the function

Thus, for the function_{C}), which, as the fundamentals of PAL2v is obtained by Equation (11):

Similarly, the Equation (12) the Contradiction Degree of the Paraconsistent Newton’s quotient:

Therefore, the Paraconsistent values extracted from Newton’s quotient adjusted to Paraconsistent Logical Model depends of

_{ψ}).

In this section it is presented examples of the use of the Paraconsistent Newton’s quotient in the resolution of first-order Derivative for different values of the Variable Increment x.

Example 1

It is wanted a final value of the first-order Paraconsistent Derivative of the function

that the Contradiction Degree of the Newton’s quotient that is near the h_{ψ}, establishes thus a Paraconsistent Log- ical State ε of equilibrium of the PAL2v-Lattice τ.

Resolution: Initially, to form Paraconsistent Newton Normalization Factor it is calculated the maximum value of the function

It is calculated the value of the Paraconsistent Newton Normalization Factor, according to the Equation (13):

For an increment value of the variable x, for example,

The Paraconsistent Newton’s quotient is calculated according to Equations (15) or (16)

Recovering value of Paraconsistent Derivative in the physical world by Equation (14):

Therefore, for these conditions, where the increment of the variable x is

Decreasing to

The Certainty Degree of Paraconsistent Newton’s quotient calculated by Equation (17) is:

And the Paraconsistent Newton’s quotient calculated according to Equation (16) is:

Recovering the value of Paraconsistent Derivative in the physical world by Equation (14) becomes:

Therefore, for these conditions, where the increment of the variable x is

Decreasing the value of

The Paraconsistent Newton’s quotient is calculated according to Equation (16):

Recovering the value of Paraconsistent Derivative in the physical world by Equation (14):

Therefore, for these conditions, where the increment of the variable x is

Decreasing the value of

The Certainty Degree of Paraconsistent Newton’s quotient is calculated by Equation (17):

The Paraconsistent Newton’s quotient is calculated according to Equation (18):

Recovering the value of Paraconsistent Derivative in the physical world by Equation (14):

Therefore, for these conditions, where the increment of the variable x is

It is verified that the Contraction Degree of Paraconsistent Newton’s quotient obtained by Equation (18) ap- proaches to

For

For

For

For

Example 2

Through Paraconsistent Mathematics originated from the Paraconsistent Logical Model, determine:

1) The equation of the first-order Paraconsistent Derivative of the function

2) Using a value of the increment of the variable x of

Resolution

1) Using Paraconsistent Newton’s quotient placed in the form of Equation (15)

where:

K_{N} is the Newton Normalization Factor, calculated by Equation (13), such that:

The Favorable Evidence Degree:

The Unfavorable Evidence Degree:

2) The first-order Paraconsistent Derivative of the function

a) Obtain the maximum value of the function at the point

b) Paraconsistent Newton Normalization Factor, will be:

c) Applying Equation (15) of Paraconsistent Newton’s quotient obtained in item a to

The value of the first-order Paraconsistent Derivative in the physical world is obtained by the Equation (14):

Therefore, for these conditions, where the increment of the variable x is

This paper presented a method for Differential Calculus using the foundations of Paraconsistent Logic applied to the Newton’s quotient. The adequacy of the Differential Calculus to the Paraconsistent Logical Model made the contradictions existing in the Differential Calculus be accepted as inherent to a model based on real situations, so an imperfect world. Thus, the Paraconsistent Differential Calculus, structured in a logic that accepts contradictions, is able to dissolve the uncertainties, aggregating values that conventionally would be discarded. Following the procedures of Paraconsistent Mathematics were presented some examples where results are obtained from resolutions of first-order Paraconsistent Derivative of some functions. The results indicate that Paraconsistent Mathematics for resolutions using first-order derivatives responds well to these basic applications and facilitates the computational treatment. In the second part of this work it will be studied the methods for using the Paraconsistent Logical Model applied to solving problems through second-order derivative.