_{1}

In this paper we show that it is possible to integrate functions with concepts and fundamentals of Paraconsistent Logic (PL). The PL is a non-classical Logic that tolerates the contradiction without trivializing its results. In several works the PL in his annotated form, called Paraconsistent logic annotated with annotation of two values (PAL2v), has presented good results in analysis of information signals. Geometric interpretations based on PAL2v-Lattice associate were obtained forms of Differential Calculus to a Paraconsistent Derivative of first and second-order functions. Now, in this paper we extend the calculations for a form of Paraconsistent Integral Calculus that can be viewed through the analysis in the PAL2v-Lattice. Despite improvements that can develop calculations in complex functions, it is verified that the use of Paraconsistent Mathematics in differential and Integral Calculus opens a promising path in researches developed for solving linear and nonlinear systems. Therefore the Paraconsistent Integral Differential Calculus can be an important tool in systems by modeling and solving problems related to Physical Sciences.

The Paraconsistent Logic (PL) belongs to the class of non-classical logics and presents in its foundation some tolerances at contradiction, without invalidating the conclusions. Its extended form called Paraconsistent Annotated Logic (PAL), has in its representation an associated Lattice that allows the development of algorithmic techniques and direct applications [

The PAL2v-Lattice [

As seen in [

where: m is Favorable evidence Degree, where

λ is Favorable evidence Degree, where

As seen in

The Contradiction Degree (D_{ct}) is obtained by:

As seen in _{ct} belong to set Â, vary on the closed interval +1 and −1 and are exposed on the vertical axis of the PAL2v-Lattice called “Axis of contradiction degrees”.

By analyzing the PAL2v-Lattice [_{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.

where: e_{τ} 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.

_{τ}.

The Certainty Degree normalized [

Likewise, the normalized Contradiction Degree from the Paraconsistent Logical Model is calculated by:

According to definition, the Derivative [

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.

The PAL treating information signals in its special form called Paraconsistent Logic with annotation of two values (PAL2v) allows extracting equations for applications in signal analysis from the Newton’s quotient.

With PAL2v applied in the Newton’s quotient, we can obtain all the information necessary and sufficient to effect the derivation of first and second order and apply them to physical systems with good results without ignoring the action of the infinitesimal [

Initially we will apply to the Newton’s quotient a factor of normalization K. This is necessary for that we can put its values within the limits of the PAL2v-Lattice, 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 (9) are identified Degrees of Evidence of PAL2v annotation, such that:

From Equation (2) we have the Certainty Degree of the Newton’s quotient:

Similarly, from the Equation (3) the Contradiction Degree of the Newton’s quotient:

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. We can adjust this value is an equilibrium point equivalent to Planck’s constant called Paraquantum Factor of quantization, as seen in [

Be

K_{N} Paraconsistent Newton Normalization Factor.

The value of the Paraconsistent Derivative of the first-order in the physical world is obtained through reapplying the Newton Normalization Factor (K_{N}) in the result obtained by 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 [_{N}) and identifies how it is represented any differentiable function f(x) before the mathematical procedure for obtaining the derivative. Thus, this normalization allows the function

where: _{C}), which, as the fundamentals of PAL2v is obtained by Equation (12):

Similarly, the Equation (13) the Contradiction Degree of the Paraconsistent Newton’s quotient:

And from the Equation (4) the Evidence Degree resulting of Paraconsistent Newton’s quotient:

And from the Equation (5) the normalized Contradiction Degree of Paraconsistent Newton’s quotient:

Calculate the final value of the first-order Paraconsistent Derivative of the function

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

The value of the Newton Normalization Factor, according to the Equation (12), is:

The Certainty Degree of Newton’s quotient is calculated by Equation (16):

The Paraconsistent Newton’s quotient is calculated according to Equation (15):

Recovering the value of Paraconsistent Derivative in the physical world by Equation (13):

Then, for these conditions of

Whereas the first-order Paraconsistent derivative is obtained with the calculation of the Paraconsistent Newton’s quotient Equation (9), then the Certainty Degree is:

This first value of the Certainty Degree will be normalized by application Equation (4), turning into Favorable Evidence Degree to the second-order Derivative, so:

Or then, (19) in (20), we have:

The equation of Paraconsistent Newton’s quotient of the second point, or second Paraconsistent logical state, obtained into PAL2v-Lattice for second-order derivative is:

Are identified in the Equation (22) the degrees of evidence, such that:

The second value of the Certainty Degree will be normalized, thus becoming by Equation (4) in Unfavorable Evidence Degree to the second-order Derivative of the same function f(x), so:

Or then, (23) in (24), we have:

For this second representation of Paraconsistent Derivative when decreases the value of

The analysis of sequence in PAL2v will result in the Certainty degree divided by the value of the square of the increase of the variable x, so:

The Equations (20) and (24) in (26), results in:

Or, making (21) and (25) in (26) and rearranging, the Paraconsistent Newton’s quotient for second-order function is:

where:

K_{N} is the Normalization Newton factor.

To recover and so obtain the Paraconsistent Derivative value for second-order function f(x) in actual physical universe:

where:

Calculate the final value of the second-order Paraconsistent Derivative of the function:

For the resolution, initially is estimated the maximum function value at the point considered

Therefore, being:

The Paraconsistent Newton Normalization Factor is calculated by Equation (12):

With the Equation (27) is obtained the second-order Paraconsistent Derivative, that with:

The value of the second-order Paraconsistent Derivative of function f(x) in actual physical universe is obtained by applying the Equation (28):

Then, for these conditions of

In the calculations of Derivative every one of the Primitive functions, called here by

The General Primitive function

By other side, the Definite Integral is as an insertion of a function and extraction a number, whose value corresponds to the area between the graph of the function and the axis of x. In the calculation of Definite Integral are established the limits of integration, so the calculation is a mathematical process established between two well-defined intervals [

The application of the concept of Integration in a function through Paraconsistent Logical Model will be made based on the Derivative process that uses the incremental rate, or Newton’s quotient. For this condition the Paraconsistent logical State from Equation (3), which is defined in the PAL2v-Lattice by the values of the degrees of evidence, is located at one point represented by the Certainty Degree (Equation (15)) and the Contradiction Degree (Equation (16)) of Paraconsistent Newton’s quotient:

This means that for any type function _{N}) is obtained with the representation of Paraconsistent logical state onto axis of the degrees of contradiction.

In the method of integration in conventional mode leads to ignore the infinitesimal, as also is made in the method of limits, when considered the increase of variable x tends to zero.

In the conventional integral method for the function

Subtracting

This equality Equation (31) compares Paraconsistent Newton’s quotient with the Derivative equation of conventional method before the increase of variable x tends to zero. To make the increase of variable x tends to zerothe term fractional on the right side of the Equation (31)

Or through another notation:

In conventional Integral method the equation of Primitive function should have adjusted their coefficient to adjust the values. Therefore, the Primitive Function of a Derivative function resulting

The value of the constant C is added to equation, thus obtaining the General Primitive function. And introducing the Indefinite Integral in symbolic mode, we have:

In conventional Derivative [

In the Paraconsistent Logic this mathematical process indicates that in the derivative is performed a contraction in the PAL2v-Lattice. Other action of applying the binomial theorem is that when it is made the Derivative; the eliminated term is corresponding to the degree of Unfavorable Evidence Degree

Therefore, for this condition, are identified:

Then, after the Derivative action, the Certainty degree, expressed by Equation (15) is:

Similarly, the Derivative action also modifies the value of the Contradiction Degree (Equation (16)), that before was expressed by:

To happen the Derivative action that nullifies the Unfavorable Evidence Degree

It is verified that the Integral Paraconsistent aims to return the Paraconsistent logical state _{ψ}). Therefore, integral action will cause the Paraconsistent Logical State

At the equilibrium point, which is under the vertical axis of PAL2v-Lattice, the Contradiction Degree has its value known, such that:

Dividing the terms of equality in the previous equation:

Therefore, the value added to the Contradiction Degree of Paraconsistent Logic State at the integral point is

After the integral action the normalized Contradiction Degree presented in Equations (5) and (18) will be:

It is verified that Paraconsistent logical state from the equilibrium point of Paraconsistent Factor of quantization

where: K_{N} is a Normalization factor of Newton, such that:

Similarly, the value of the constant C is added to the Equation (38), thus obtaining the Primitive function General:

Multiplies the value of K_{N} to the result obtained in the PAL2v-Lattice, and the Primitive function final will be given by:

The Paraconsistent Integral Undefined is presented in symbolic mode, such that:

Thus, the calculation of the area will be:

1) For the area in the second point of the curve x = b:

2) For the area at the first point of the curve x = a:

The total area is calculated by:

Consider as a first example that is given the Derivative function

Resolution: Initially, n appears in the Derivative function as:

Using the Equation (39) the Primitive function by paraconsistent mode will be found:

As second example considers that given a Derivative function from the type

Resolution: Note that the Derivative function

Consider as a third example where we use a Paraconsistent Integral Calculus to determine the area under curve

Resolution: In the resolution we can use the Equation (41) with an Increment value of variable x of

For

Resulting:

For

Resulting:

Area calculation by the equation (42):

Consider another example where is used the Paraconsistent Integral Calculus to determine the area under the curve

Resolution: In the resolution, using the Equation (41) we can consider an Increment value of variable x:

For

Resulting:

For

Resulting:

With the subtraction of areas using Equation (42), we have:

As the example 5 consider that using an increment value of the variable x of:

Resolution: The resolution is done using the Equation (41):

For

For

We found the result of the area using the Equation (42):

Resulting:

This article presented a Paraconsistent Mathematics that structures a method for differential and Integral Calculus using the foundations of Paraconsistent Logic applied to Newton’s quotient. The study allowed an adequacy of Differential Calculus to Paraconsistent logical model. With this, existing contradictions are accepted as inherent to a logical model based on real situations, therefore of an imperfect world. It was found that the Differential Calculus, structured in a Paraconsistent Logic that accepts contradictions, is able to dissolve the uncertainties, adding values that conventionally would be despised. Even requiring further testing involving more complex math functions the results obtained are very promising and suggest good perspectives for future applications of differential and Integral Paraconsistent Calculus.