Relativity Theory and Paraquantum Logic — Part I : The Time and Space in the Paraquantum Logical Model

From fundamental concepts of the Paraconsistent Annotated Logic with annotation of two values (PAL2v), whose main feature is to be capable of treating contradictory information, was created the Paraquantum Logic (PQL). The studies of the PQL are based on propagation of Paraquantum logical states ψ in a representative Lattice of four vertices. Based in interpretations that consider resulting information of measurements in physical systems, are found two Paraquantum factors: the Paraquantum Gamma Factor P  , that has his action in the measurements of Observable Variables in the Physical world and the Paraquantum Factor of quantization hψ, which has his action in the Paraquantum World represented by the PQL Lattice. Correlation between P  and hψ produces paraquantum equations for computation of the physical quantities in real physical systems. In this work we present a study of application of the PQL in resolution of phenomena of physical systems that involve concepts of the Relativity Theory. Initially the time t is considered like an Observable Variable and the paraquantum analysis is done with the same conditions assumed in the relativity theory for the study of the time dilatation. After the time considerations, paraquantum equations are involved with the space-time and velocity creating conditions for a relativistic/paraquantum analysis. In the part II of this work a new approaches of the relativistic phenomena in the Paraquantum Logical Model will show the correlation of these effects with the Newtonian universe and with quantum mechanics.


Introduction
A Paraconsistent Logic (PL) is a non-classical logic which revokes the principle of non-Contradiction and admits the treatment of contradictory information in its theoretical structure [1][2][3].The Paraconsistent Annotated Logics with annotation of two values (PAL2v) is a class of Paraconsistent Logics particularly represented through a Lattice of four vertices (see [4]).
Under certain conditions the results obtained from the LPA2v model changed through leaps or unexpected variations [5].With that it is verified that the application of its foundations presents results strongly connected to the ones found in modeling of phenomena studied in quantum mechanics (see [5][6][7]).Because of this behavior, with fundamental concepts of the Paraconsistent Annotated Logic with annotation of two values (PAL2v), was created the Paraquantum Logic (P QL ).Through the paraquantum equations we investigate the effects of balancing of Energies and the quantization and transience properties of the Paraquantum Logical Model in real Phy-sical Systems [5,8,9].
We can obtain through the PAL2v a representation of how the annotations or evidences express the knowledge about a certain proposition P [4].This is done through a lattice on the real plane with pairs  ,    which are the annotations.If P is a basic formula then: where ~ is the logic negation of P and ,   0,1 Considering that μ is the favorable degree of evidence and λ is unfavorable degree of evidence, then the symbol   , P   can be read in the following way: With the values of x and y that vary between 0 and 1 and being considered in an Unitary Square on the Cartesian Plane (USCP) we can get linear transformations for a Lattice k of analogous values to the associated Lattice τ of the PAL2v [4,5].The obtained final transformation is: According to the language of the PAL2v, we have: x   → is the favorable Evidence Degree; y   → is the unfavorable Evidence Degree.The first coordinate of the transformation (1) is called Certainty Degree (D C ) and is obtained by: ( The second coordinate of the transformation (1) is and is obtained by: The second coordinate is a real number in the closed interval   1, 1   .The y-axis of the Lattice τ is called "axis of the contradiction degrees".From (1)-(3) we can represent a Paraconsistent logical state ( τ ) into Lattice τ of the PAL2v [4,5], such that: D ct is the Contradiction Degree obtained from the evidence Degrees μ and λ.
Figure 1 shows sequences for obtaining the Lattice of Paraconsistent Annotated Logic with values.

The Paraquantum Logic (P QL )
With base in the concepts of the LPA2v the foundations of the denominated Paraquantum Logic (P QL ) are considered (see [4,5]).
Initially it is verified that the Equations ( 2) and (3) can be expressed in terms of μ and λ obtaining: Then, a Paraquantum logical state ψ is created on the lattice of the P QL as the tuple formed by the Certainty degree (D C ) and the Contradiction degree ( Both values depend on the measurements perfomed on the Observable Variables in the physical environment which are represented by μ and λ [5, 8,9].
A Paraquantum function is defined as the Paraquantum logical state : For each measurement performed in the physical world of μ and λ, we obtain a unique duple which represents a unique Paraquantum logical state ψ which is a point of the lattice of the P QL [5,8].
On the vertical y-axis of contradictory degrees, the two extreme real Paraquantum logical states are: 1) The contradictory extreme Paraquantum logical state which represents Inconsistency T: 2) The contradictory extreme Paraquantum logical treme real Paraquantum logical states are: 1) The real extreme Paraquantum logical s presents Veracity t: presents Falsity F: will have origin in one of the two vertexes that co e the horizontal axis of the certainty degrees and its extremity will be in the point formed for the pair indicated by the Paraquantum function where: D ct = Contradiction Degree computed by (4).

Usin ule of a V
D C = Certainty Degree computed by (5).g (6) which is for computing the mod ector of State   P  , we have: 1) For D C > eal Certaint 0 the r y Degree is computed by: Therefore: gree is computed by (4); by (5).
2) Fo 0, the real Certainty De by: Therefore: 0, then the real Certainty Degree is nil: is computed by: The intensity of the real Paraquantum logical state The inclinat ion angle   of the Vector of State which is the angle formed by the Vector of State   P  and computed by: the x-axis of the certainty degrees is

Uncertainty Paraquantum Re
When the module of the So, the Paraquantum Factor of quantization in its complete or total form which acts on the quantities is:

Newton Gamma Factor
Comparisons and analogies between the International unit Systems and the British System result in a proportionality factor related to the British system [8,10].
br Given the importance of the Factor br , which will be largely used in the equations of the P QL , its value is called Newton Gamma Factor whose symbol is Therefore, in order to apply classical logics in the Paraquantum Logical model [8,[10][11][12], the Newton Gamma Factor is 2 . take ntum L

Paraquantum Gamma Factor Pψ γ
For an expansion process of the P QL Lattice where we , consider quantizations based in consecutive applications of inversed values of the Newton Gamma Factor we can identify the Lorentz Factor  in the infinite Power Series of the binomial expansion [8,10].
In the paraquantum analysis [8] related to the series obtained from consecutively applying the Newton Gamma Factor N  Gamm we define a correlation value called Paraquantum a Factor   such that: where: N  is the Newton Gamma Factor:

The Study of the Time Flow in the Paraquantum Analysis
In the Paraquantum analysis applied to Physical Systems, time must be considered a an Observable Variable whose measurements must be inserted iscourse (or Interval of Interest) from where the Evidence Degree ( or   ) is extracted from.However, for the analysis in Paraquantum Logical Model, when compared to other fundament hysical quan the al p tities, time has interesting features.Among these, measurements of the Observable Variable time have ties: er decre ake es; 2) Its v never constant 3) Measurements are mandatorily quantized due to its fluidity feature.

The Measured Time t Considered as an Observable Variable
The study of time as an Observable Variable in paraquantum analysis can be done from the same conditions assumed in the relativity theory for the study of the time dilatation [10,13].
First, we consider the classical example from the relativity theory where an observer O' at rest on the referential S' at a distance L from a mirror, according to Figure 3(a).The observer sends a light pulse through Transmitter E and measures the time interval between the emission instant and the return instant of the pulse reflected by the mirror M. Since light travels with velocity c, we can compute this time interval by: the following proper-1) Its value always increases and nev ases which m time to be a flow of positive valu alue is never stable, therefore, is ; t .
We now consider both events: one is the transmissio of the light pulse by E and the other is its return to th r O' is but now er for referential S than for referential S'.This new condition arises because, since v was brought up, for the observer at referential S there is the horizontal distance n e point where observe observed from other referential S. In this second condition of the referential S, according to Figure 3(b), observer O' and mirror M are moving to the right with scalar velocity v whose value is a fraction of the velocity of light c in vacuum.For the observer at referential S, both events now happen in different places, such that: Event 1 → Emission of light pulse by E in X 1 ; Event 2 → Return to the observer O' in X 2 .
According to the figures, the space traveled by the light pulse is larg is the measured time interval at referential S. According to the relativity theory [10,13], the value of the velocity of light in vacuum is a constant c, with  respect to the referential S as well as with respe referential S'.Since the sp ct to the ace traveled with respect to the referential S is longer, then when compared to the time passed with respect to S', the time spent for the light pulse emitted by E to hit the mirror M and to return to the observer O' is also longer.It is seen on Figure 3(b) that the course of the light pulse observed from the referential S generates a rectangle triangle of vertices E X1 -E X -M.We can use the Theorem of Pythagoras on this rectangle triangle to analyze the situation [10,13]: Isolating L 2 : We do: By inverting, we obtain: where: t  = Variation of the total time with respect with the referential S. t  = Variation of time with respect to the referential S' when measured at the referential S. In quantitative terms it means that to each unitary variation of the quantity of time measured at S ( 1 t   ), there is a decrease of the value of variation of measure time t  related to the body which has velocity v.We call decresc t  the decreased variation of time and we have: Considering the example for null velocity, And for velocity value: Being me featured with a flow that always increases, wh ti mulate the following: en it is considered as one of the Observable Variables from where the Evidence Degree (μ) is extracted, it shows this property in quantitative terms.We can for- where: to the time fraction will be increased t the time measured at the referential .So, in each unit of time measured at S, there will be a corresponding value decresc t  t will be that will be This quantity tha a to measured at the referential S is gi added.the quant by: dded ven ity of time that acts of time when this one is re-on the amount : Doing so, for the variation of the qua ntity of total time to be unita ured at the referential of the Universe of Discourse is considered in the paraquantum analysis together with the se lue of the Factor of Lorentz.
ry, the quantity of the variation of time measinver va In the Equation ( 22) the following implication exists: In these conditions were Quantity of total time is unitary, the Quantity of time measured at the referential of the Universe of Discourse of the paraquantum analysis is:   4) and ( 5) are now dependent of the time measurement, therefore, they become:

The
And the Paraquan ical state (): tum log


We can verify that the t as action direct measurements of the Observable Variables of the physicca

W
e fundamental laws of physics [10][11][12], in the analysis of acceleration related physical quantities: one is time and th ime h ly in the l world.The variation of the time makes the values of the measurements in the Observable Variables modify and, as consequence, appear a propagation of the Paraquantum logical state through the P QL Lattice.quan e can study the flow of the Paraquantum time, having in mind that, according to th there are two the other is e velocity v, whose value is contained in the attenuation Lorentz factor  in the relativity theory.Since, in the paraquantum analysis, there is the need of two evidence degrees to form the annotation (which are extracted from the physical quantities considered as Observable Variables), then, for a dynamical process, where the condition of acceleration is valid, there will be the the Evidence Degree from the Observable Varia A O Quantity of total time.Since the Observable Variable time has the feature of being a flow that never stops and always increases, for its measured value to be contained in the closed real interval [0,1], then from Equation (22): need to extract ble time.s it is done in the other physical quantities, the extraction of the Evidence Degree for the Observable Variable time will produce a normalized value in the closed real interval [0,1].For the unitary value of the quantity of total time at the referential of the Universe of Discourse of the paraquantum analysis, the Evidence Degree of the bservable Variable time must be a fraction of unitary total 1 Q t    → Universe of Discourse is unitary and from Equation (23): We can consider the time measured of the Universe of Discourse of the par on (21) the at the referential aquantum analysis in Equati favorable Evidence Degree of the Variable Observable time ( Δtime  So, the greater the Factor of Lorentz  is, the closer to zero is the unfavorable Evidence Degree extracted from the Observable Variable time that is.

The Lorentz Factor in the Quantization of the Paraquantum Time
According to the concepts of the P QL [5,8,9] we observe the correlation among the values extracted from the Observable Variables of the physical env form of Evidence Degrees, that leads to an equilibrium situation, defined by a paraquantum lo lished by the Paraquantum Factor of quantization h ψ .So, th Factor of Loren Paraquantum Logical Model.For the existence of the correlation, we must compute the Factor of Lorentz ironment, in the gical value estabere must be an adaptation for using the tz in the equilibrium condition of the  considering that for the equilibrium condition, the value corresponding to the Contradiction Degree on the vertical axis of the P QL Lattice is the Paraquantum Factor of quantization h  .This procedure is done by starting with the value of the favorable Evidence Degree extracted from the Observable Variable time with the dependency of the Factor of Lorentz for the equilibrium condition.
The involved values in the equilibrium condition can gree be represented on the equation of the Contradiction De (4) such that: For equilibrium point on the contradiction degrees y-axis:    .Therefore: The unfavora le Evidence Degree and the Lorentz Factor when being affected of the paraquantum inverse value, such that on Equation ( 27 Through this equation, we can obtain the Factor of Lorentz for this equilibrium condition: If in the study of Relativity Theory the velocity v is the Observable Variable for extraction of favorable Evidence Degree (μ), then the time can be used as Observable Variable for extraction of the unfavorable Evidence Degree (λ).
The value of velocity v related to the value of the speed of light c in the vacuum in the equilibrium condition, can be obtained from the value of the Factor of Lorentz  , such that:   equality considered at the equilibrium condition, we have: Therefore, paraquantum analysis applied i theory the favorable Evidence Degree extracted from the Observable Variable velocity v related to the speed of Li is n the relativity ght c in the vacuum, in the equilibrium condition, : And the unfavorable Evidence Degree is: Δtime 1 2   .

Adjust of the Lorentz Factor in the Paraquantum Time
For the corresponding values of the Evidence Degrees extracted from the equilibrium condition [5, 8,9], the an e alysis on the P QL Lattice indicates on th vertical y-axis of the contradiction degrees the Paraquantum Factor of quantization h ψ , such that: 2 1 h    .We can consider that a certain X paraquantum value is en the v easure by the analy-the responsible for establishing the connection betwe alues obtained from the time m sis of the restricted relativity theory and the quantization on the P QL Lattice.Through the equation of the Lorentz Factor, we observe that this can not be less than 1, therefore, this means that there is a certain paraquantum in- acting on the Factor of Lorentz  .
This inverse paraquantum value adjusts the Evidence De- , which means that, at each measurement of the equilibrium condition, the quantization is done by the Paraquantum Factor of quantization h  represented on the vertical y-axis of the contradiction degrees of the P QL Lattice.From Equation (27) action of the inverse value the 1 X is: 1 The Contradictio e in the equilib n Degre rium point by Equation ( 4) is: → where we get the X value by: 1 1 2 Including the Newton Gamma Factor, such that:

 
For the unitary Factor of Lorentz, therefore null velocity or close to null, as in the Newtonian universe, the resulting Paraquantum Gamma Factor is: When the Paraquantum Gamma Factor is e r of Lorentz will be computed as follow unitary, th Facto s: The value of velocity v related to the value of the speed of light c in the vacuum in this condition, can be obtained from the value of the Factor of Lorentz  , such that: 0.52100538 velocity 3c   .The Paraquantum Gamma Fac r of Equation ( 17) is the same one that appears on Equation (28) and its goal is to correlate the values of measurements in the physical world with the logical states in the paraquantum world.

The Space-Time in Paraquantum Logic Analysis
to l be n sidered in the analysis, then it would not be a Paraquandiction.For 7) in the Cont If the time wil the only physical largeness to be cotum analysis, but classic.This happens because it will not have contra example considers the Equations ( 26) and ( 2radiction Degree with total 1 Q t    computed by Equation (4): The Certainty Degree computed by Equation ( 5) is: And the Paraqua logical state ) of the time is: If the an 1, then, in th the E 8) alysis is done using the Lorentz factor equal to accordance wi quation (2 , the value of Paraquantum Gamma Factor is less than 1. This f Paraquan amma Factor less than 1 removes in the Equations ( 26) and ( 27) the normalization status of evidence degrees: Therefore, Δtime This characteristic of the en considered as O characteristic inversely proportional is regarded as a flow, which starts from its maximum unitary value.
For the real world the values of evidence degrees of tim and (27) for various val velocity hown in Table 1 e value of the Degree of Contradiction will always be null, independently of Paraquantum Gam Factor value.Therefore, using only a single source of information (space or time) the equilibrium point of the Paraquantum lo It is verified that for a single source of information, th ma time, wh bservable Variable, demonstrates the natural action of his always to act in the sense of expanding the Lattice.
In the representation of the time the gical state (ψ), that is established in the y-axis of the contradiction degrees for 2 1 e related to the Lorentz factor in the relativity theory can be calculated by Equations (4) (5) (11) (26) ues of v, as s .

The Space/Time
Extraction of the Evidence Degree as Source of Information for In Physical Systems Analysis the value of velocity (v measured ) velocity values using the paraquantum rel ti  Based in Equation (23) the variation of measured time t  is considered by the following equality: For the paraquantum analysis this equality means that the total amount of time variation , which defines the iverse of dis u by the Paraquantum Gamma Factor.
For the condition of velocity, the relationship between the values of favorable Evidence e sp (which doesn't have the influence of Paraquantum Gamma Factor in their measurements) and of the time (which has the influ ce of Ga in th measurements) is: When the analysis is made in the universe of the relativity theory related to the speed the light in t e vacu fore, ition factor of Lorentz And th ree is com ted by Equati n (5), were: From Equation (32): he P Gam be us When: From Equation (31): From Equation (32): From Equation (31): e relativity theory.These results show that exist in paraquantum analysis applied in the theory of relativity a single point of correlation.

The Equilibrium Point in Paraquantum Analysis
According to the results in Table 2 presented by paraquantum/relativistic analysis verifies that exist an equilibrium point in the P QL Lattice.The equilibrium point that relates to application of Paraquantum Logic in the various areas of hysics, including the theory of relativity, is axis of the d

Discussion
The Evidence degree values of space/time an ervable so, are dep dent of the Lorentz tum analysis of th Physical Syste e Newtonian universe the sp as Variable Observable, and the time as le Observab , are considered separately.The equations of paraquantum veloc e, work and energy are extracted from Newton's laws and in paraquantum analysis consider calculation always in equilibrium point located in the vertical y-axis of the P QL Lattice.
However, in the Newtonian universe the velocity in   , but obtained by dividing the space and time, where only time suffers the action of Paraquantum erse the E ree of Observable Variable velocity is equations extracted in the Newtonian universe, is always the inverse of the factor of Newton: these equations is not related to the speed of light in a vacuum c 1 1 2 This value in the Physical world marks the variations of un Gamma Factor.This manner, in the Newtonian univ vidence Deg certainties that happen within the P QL lattice in quanation form of the axis of y extracted of two source of information the space and the time.As, for these equations the velocity that is related to tiz modifying the intensity of the Paraquantum logical State.
to calculate energy levels in hydrogen atom.
contradiction degrees b the speed of light in vacuum is equal to zero, then the Lorentz factor is unitary ( 1   ).This causes the value of This quantization can be expressed in the form of representation of energy, as it was studied in [14] and [15] the Paraquantum Gamma Factor, which acts in these  The effects related to energy in the Newtonian universe, in the universe of the theory of relativity and in quantum mechanics resen gle L tice of the Paraquantum logic.In the part II of this work we will study these

Conclusions
In this paper we pr dy o no ena that correlates the concepts of the Theory of the Relativity and fou the m lo (P QL ).Through the originated Equations of the Paraquantum Logical Model we did analogies with relativistic effects where we ve lation e Fact Lorentz γ and the Paraquantum Gamma Factor The effects of these factors were studied in detail and we observed that the Paraquantum Gamma Factor P  , which aggregates the phenomena found in the theory of relativity and in the Newtonian universe, promotes the connection among the physical universes through the correlation with the Paraquantum Factor of quantization h  .Through the paraquantum equations we can consider time as an Observable Variable and thus extract degrees of evidence for the analysis.It was studied by paraquantum analysis as is the action of the time in the theory of relativity which can only be considered as a single factor of space/time.We studied as time, space and velocity can be correlated in relativistic and Newtonian world via the logical state.D C is the Certainty Degree obtained from the evidence Degrees μ and λ.

2 1Figure 1 .
Figure 1.Sequences of linear transformations for obtaining the Lattice of Paraconsistent logic with values.

FY
The Vector of State   P  will always be the vector addition of its two comp vectors:C onent X Vector with same direction as the axis of the certainty degrees (horizontal) whose module is the complement of the intensity of the certainty degree:1 Vector with same direction as the ax -is of the con tr ntum logical state ψ cur defined by adiction degrees (vertical) whose module is the contradiction degree:

μ λ ct μ λ D D Figure 2 .
(a) Paraquantum logical state (ψ) into the paraquantum lattice on the point ; (b) The paraquantum factor of quantization h related to the evidence degrees obtained in the measurements of the observable variables in the physical world.
the Lorentz factor which is: Using the Paraquantum Gamma Factor P  allows the computations, which correlate values o servable Variables to the values related to quantization throu s in a Universe of D f Ob gh the Paraquantum Factor of quantization h ψ [5,8,9].

Figure 3 .
Figure 3. Analysis of the observable variable time (Similar an ity theory).
(b) alysis to the study of the time dilatation in the restricted relativ-Copyright © 2012 SciRes.JMP J. I. DA SILVA FILHO 962 time variation-Factor of Lorentz  .The last expression from uation (18) indicates the relation of the variation of measured time or local time at th when this one is unitary with the time measured at the referential S'.So, we can establish a quantitative relation between the time measured t the unitary value () and the Factor of Lorentz Eq velocity v of the body being considered.

=
Quantity of time measured at the refv = 0; Added Q t  the referential S due t = Quantity of time m o relativistic effects.Therefore, for the observer at th tial S quantity of the time variation that dec easured and added e referen reases in to , the t  will o be the same quantity of variation that S measured dded will be the com Factor of Lore locity  (20) indicates that when at th er the quantity of time a plement of the inverse value of the ntz.Considering the example for null ve .
time measured at the referential of the Universe of Discourse of the paraquantum analysis.We observe that in the ph world the tim antization in the paraquantum analysis is related to inverse va Consideri m of a Pa onsistent logic with annotation of two (PAL2v ,5,8], the concep f Pa ysical e qu lue of the Lorentz Factor. the fundamentals of Paraconsistent Logic (PAL2v).Thus, the values of the Certainty  

Figure
Figure 4. Re esentat ces d he ac e flo ua con lativity theory.Propagation of paraquantum logical State (ψ) as the D C and D ct values of Table 2.
Figure 4. Re esentat ces d he ac e flo ua con lativity theory.Propagation of paraquantum logical State (ψ) as the D C and D ct values of Table 2.
equilibrium point in Paraquantum Logic Model.As demonstrated in this work, the results are easy to view ugh physical science.The paraquantum equations used in e are d pos the of unified calculation for Physical Science.

forms an angle of 45˚ with the horizontal axis of certainty degrees. The unbalanced contradictory Paraquantum logical state ctru is the one located on the lattice of states of the P QL where there is a condition of opposite signs between th
P 

Time and the Paraquantum Logic
then modified 

Table 2
point of equilibrium is located into P QL Lattice the Paraquantum Factor of quantization h  exactly where we can use either the Paraquantum a Factor Gamm