Chapter 3 (The Sign Rules of Opposites)

Qualitative Rules
   The qualitative rules (specifically sign rules) are those that constitute a consistent set of relationships. It is like the multiplication table, which is considered a set of systematic rules satisfying each other, where its symbolic consistency is held during the substitution process from one rule into the other. The multiplication table can be considered consistent symbolic relationships without any relevance to multiplication, division, or even any numeric meanings; what concerns us is its consistency.

   To the best of my knowledge, all sciences are devoid of consistent relationships except the physical laws and the known sign rules, moreover the multiplication table. If I'm not mistaken there no much more, even Boolean algebra can't be included because of the lack of substitution relationships. Despite the qualitative rules are being qualitative in nature, this doesn't mean they exclude quantities; because quantities go underneath quality, because quality is the most global and general concept.

   Considering the Euclidian geometry and its beauty, and its gradual building upon axioms and postulates, this edifice does not represent a symbolic closed structure despite its logical consistency, unlike the physical laws that represent a consistent symbolic logic through substitution, and it is supposed to be closed on itself even if we have not attained or could not see this closing stages somehow.

   It is required with qualitative rules to use symbols, and the consistency ought to be achieved through non contradictory substitution, and it is not mandatory (only for the time being) to be closed on itself, so we find the multiplication table is endless, and its relationships goes forever without contradictions.

   The importance of qualitative rules emerges from its representation for formal general relationships relating to physical facts; and not relating to the formal thought as it has been in traditional logic science with its concepts of positive and negative predicates. Qualitative rules are general molds that include all the other relationships; they are general models that unifies between phenomena. Upon these qualitative relationships, their deduction, and their meanings; we can construct a complete science – where they represent the tools that the theory of opposites depends upon to study and unify all phenomena.

The First Set of Sign Rules
   For easy reference later on, we will call the known sign rules as the first set of rules; which are:
+ x + = +     (1)     - x - = +       (2)
+ x - = -       (3)     - x + = -       (4)

   Since these rules represent the relationship between two states (+,-) then the second set of rules governs the relationships between three states (+, -, ±). Thereby, the number of the set is lower by one than the number of states it includes, namely the first set includes two states, and the second set includes three states and so on.

The deduction of the first set as it is known goes as follows:
The deduction starts with some assumptions, like considering gain or profit is positive, and loss is negative, likewise with the pairs: (after, before) – (above ground level, under ground) – (increase, decrease). It is all positive and negative states respectively. Then the deduction continues with mentioning some examples from daily life such as:
If a person gains (+) 500 pounds monthly, find his income after (+) three months? Sure the result should be positive, because the person would earn, thus: +500 x +3 = +1500, so we deduce: +x+=+, and the proof goes on to deduce the rest of the rules, but the objections against this deduction are:
The basic assumption considers "gain" and "after" as positive, and "loss" and "before" as negative, but this is unacceptable assumption, because these are really two assumption not one. If the assumption started with "gain" as positive, and "loss" as negative and then tried to drive the state of both "before" and "after" it would be more acceptable, but this is not what goes on. It is in the first assumptions hasn't indicated any relationship between them, so their signs are completely random and unjustifiable.

  The other weak point in this deduction, overlooking the wrong assumptions; it is not a general proof. What makes us sure that when we change concepts like "before and after" or "gain and loss" with any other concepts or categories; it is guaranteed to be still applicable and valid and be compatible with all meanings?!!

   The ambiguity reaches its extreme when proving the rule: – x - =+, which is described as the most slippery or tricky one. If one person loses (-) every month 100 pounds, then what is his earning (+) since (-) three months? This is because "loss since a period of time" is considered gain with respect to the current situation, so the previous situation is better than now, and thereby it is considered as gain. Moreover, the deficiency comes to its peek when determining whether the result is gain or loss in the header of the problem itself, thus it is assumed before deducing it.

   The relatively better method (just for the time being of explanation) is to deduce these rules by following the trial and error method as follows:
Suppose that: + x + = + (1) ,   +=+/+
Then suppose: + x - = + ,  -=+/+ but this contradicts with (1) because that way: +/+ gives two different results, then we have to change our second assumption to be: + x - = - (2) ,  +=-/- which is ok.

   Then suppose: –x-=- ,  -=-/- but this contradicts with (2), thus we have to change it to be: -x-=+ (3).

   That way we have deduced three consistent rules; which what we call the first set of sign rules, where the second rule does not contradict with the first, and the third does not contradict with the second, thereby the third does contradict with the first. All right, but this not also the actual convincing proof – matters will be elucidated as we go on – whatever it is, it is better than the known proof and it is general, because it is irrelevant to certain meaning or category.

   The method used, namely trial and error will be used to deduce the relationships between states (+, -, ±), but for much more states this method is very complicated if we use the manual work, so a computer program has been developed for this purpose, where its logic is really very interesting and has remarkable consequences, and sure not depending on trial and error, but it is established actually on all what is mentioned here of simple facts about states.

The first set of sign rules can be reformulated in a reverse form which is:
- x - = - , + x + = - , - x + = +
These reverse rules of the first set can replace the first set without absence of any meaning, except as if we considered gain is negative and loss is positive, and it can be deduced starting from the assumption: –x-=- , and continuing with the same steps as the previous deduction by following trial and error method.

   The interpretation of the first set of rules can be seen as if the similarities have output that is reverse to the different states output (similarities are +x+=+ , -x-=+ , different states are: +x-=- or according to the reverse set similarities are: -x-=-, +x+=- and different states are: +x-=+) if the output of similarities is positive, the output of different is negative and vise versa. The first set has two forms and there is no preference of one over the other.

   The known sign rules according to what is known implies quantitative meanings which can be more elucidated better from the known axes system (Cartesian system), that divides the plane into four divisions, considering what is above the origin is positive, and what is below is negative, and what is to the right is positive, and what is to the left is negative, thus the increase (+) for both positive or negative means going away from the horizontal or vertical axis (either up with positive or down with negative, or right with positive or left with negative) thus the state remains unchanged when moving away, and hence +x-=- , +x+=+ (increase of negative is negative, and increase of positive is positive), whereas in the case of decreasing (-) any of them, it goes towards the reverse direction to the other state, thus the decrease of positive is negative and the decrease of negative is positive namely: -x+=- , -x-=+ , and this may be the best generic meaning of the first set of sign rules, despite the axes system in based on it, but it is not a means for its deduction, but just for illustration (it is actually build on these meanings, and it is not derived from it). Notwithstanding it will be evident later on that the first set has no quantitative meanings at all.

   Thereby, if the first set has a quantitative meaning that ensues to a qualitative change, so why there is no neutral state, since the availability of the positive and negative imposes the presence of a neutral state, which is absent in these rules? The simplest issue is that the basic definition of opposites is impossible without its availability (since the two opposites are not defined by each other, but as a function of the neutral state). The worse is that the first set has a reverse form, and the use of any of them depends upon the first assumption of states.

   These deficiencies will not be there in the second set that includes (+, -, ±) where it does not depend on the first assumption of states whatever the assumption is; it remains the same, and the reason as will be explained is that it contains the original rules and its reverse rules at the same time, which means that it is self complemented, and signifies its stability and self completeness. Moreover, the traditional sign rules deals with white and black, consequently the absence of the intermediate states. The whole being is not white and non-white (which is a blind vision) considering only the positive and negative, means the absence of the interaction between them which accrues to the neutral state.

   The first set of rules are not self complemented, and the symmetry in rules should exist, where each rule must have a reverse rule, but the first set does not include compound states, so the existence of two states must conduce to the emergence of new states.

The Second Set of Sign Rules
   The second set is that set which includes the relationship between three states: (+, -, ±) that can be deduced by following the trial and error method as follows:
Suppose that the ratio: +/- has a result +
Then suppose that the ratio: -/+ has a result -
From first assumption: +/-=+ , \ -x+=+
From second assumption: -/+=-, \-x + = -
It is clear that there is a contradiction between them, since the left side is the same, but with different results at the right side, so we may reverse both of the assumptions to be:
+/-=- , \- x - = +                     (1)
-/+=+,   \ + x + = -                 (2)
Then suppose: ±/-=-, which is rejected because from (1): -x-=+ not ±, consequently we have to reverse our assumption to be:
±/-=+ , \     - x + = ±               (3)
Now it easy to deduce:
± x - = + x – x - = + x + = -      (4)
± x + = + x – x + = - x - = +     (5)
± x ± = - x + x – x + = + x - = ±         (6)

   These six sign rules constitute the second set (The standard second set), and it represents a consistent set, whereby the substitution from any one into the other results in two equal sides.

   What we said is only one way of unjustified reasoning which can branch in many different ways, and can start from many other assumptions. Whatever it is, we will get the same rules but in different forms. So, let us go in another reasoning just of elucidation.

   Instead of reversing the first two previous assumptions, let us reverse only one of them, supposedly the second one to be -/+=+, hence we get two non-contradictory rules:
+ x - = + (1)
+ x + = - (2)
(Which looks like the revered first set)
Then suppose: ±/-=-, which is ok with the two previous rules, thus:
- x - = ± (3)
If we suppose that ±/+=+ or ±/+=- both is rejected because they contradict with the previous rules, then ±/±=±, and we get:
± x ± = ±              (4)
+/±=+ , + x ±= + (5)
-/± = - , - x ± = -  (6)

   Astonishingly, It is really the same set with keeping the neutral state as it is, but exchanging + and – with each other. So forget it completely now to avoid confusion till you tighten your grip on the basic concepts and matters will be more clear later on.

   We can follow many other reasoning of trial and error and we can get 6 forms of this second set (I just wanted to make you sure about our discussion, but it is not the suitable time at all). To avoid confusion with the reader, I'll adhere to the standard set, because their meanings are easier to understand.

   Science is devoid of any symbolic sign rules closed on itself except the first set of rules and the physical laws (this to the best of my knowledge). As the number of states increase; its ability to handle more structured phenomena increases too, the matter is not like solving equations, but it is just looking for general relationships that control the whole being- it is more general than any law, because it is emptied from any content and formulated symbolically to rule the building of the whole universe. It is logical rules applicable to any system that includes a finite number of states in a logical consistent relationship. Can we imagine that the unification of the human knowledge resides in very simple principles?!!

   All sciences do not include any consistent relationships except the physical laws and the relationships between numbers (the multiplication table), even numbers relations is assumed to be just an assumption with no justifiable logic, it is only a different symbolic representation between the right and left sides as we may say: 2+2=4 it is tautology and does not say much more that the left side is the same as the right but in different figure in an ordered systematic form.

   The connotation of these rules and what issues it can come up with; need a through investigation and poses many issues like:
* Are sign rules can be deduced for any number of states, or it has a certain limit, and if so, or not, why?
* What is the general method to deduce the rules between any number of states?
* Is it necessary for the number of states to be odd to have a final neutral state with no opposite?
* What the benefit of higher sets, and does it include the lower sets?
* Are the formal and the actual physical relationships both can be governed by only one set? And if not, so, is there is some transformation to unify them all underneath one?
* Are there certain categories from which the sign rules can be deduced (like the issue of induction – and to what extent it applies to all the other observations) or the reverse is true, so sign rules is the base on which meanings are built upon?
* In any consistent set, is there is a minimum basic number required of rules to establish the remaining ones?

   If you think about the Yin, Yang symbol, and how both of the two opposites contain each other, where the black half contains a white dot and vise versa. It is very easy to recognize that the first set of sign rules cannot realize that, where this symbol contains symmetry; and each of the two opposites contains the other and defined by it, which all is very obvious from the second set of sign rules (+x+=-, -x-=+). Intuition and common sense are dominating; and both reflect the deep simplicity of the truth about our world; and till the end of the material presented; this all will be assured within every step. 

   Endless questions continue to arise to constitute a new science, and to open the door for investigation in a new world of thought.

The Explanatory Proof of the Second Set
   From the neutral position and what it expresses about of equal homogeneity … if it is divided into two parts, and one unit is transferred from one side to the other, then one of them is greater relative to equality by one, and the other is  smaller relative to equality by one, thus the difference between the two opposites is two units (4=4, 5=3, 5-3=2, also: 3=-, 4=±, 5=+). Hence the relativity of each unit with respect to neutrality is the opposition; and with the same magnitude. But is the opposition will still be present between the two opposites when overlooking the neutral?

   The value 5 is increased from 4 by the same value that 3 in decreased from 4, the relationship is reversal in the two directions, is there any doubt about that?! The relation of + with – ought to be opposite in comparison to the relation of – with +, thereby the relation is essentially opposite in the two directions between any two states. This is a very elementary and basic principle.

   Suppose two states s1, s2 then if s1>s2, then s2<s1, in other words: if s1/s2>1, then s2/s1<1. Thus "one" is the relative reference for opposition, and with neutrality, unity (oneness) is attained. (It is a very simple and intuitive principle that is not asserted on; in spite of its importance and complete clarity). It can be said that the matter is not a quantitative comparison at all, but it is basically a relational order, where it can be said again far away from quantification, considering two points x, y: x is to the right of y, whereas y is to the left of x, thus the relative relationship is definitely opposite in the two directions.

   It can be said that the matter is not a quantitative comparison at all, but it is basically a relational order, where it can be said far away from quantitative concepts, that the relationship can express about an action somehow, so that the effect of hot body on another cold one is not the same of the cold on the hot, since one gets hotter and the other gets colder; apart from any quantitative meanings

   The very simple numeric relationship between 3, 4, and 5 indicates that: neutral relative to positive is smaller (±/+=-) and negative relative to neutral is smaller (-/±=-) and positive relative to neutral is greater (+/±=+) and the neutral relative to negative is greater (±/-=+). Now replace the words greater and smaller with right and left, and you will find that the same logic applies. So, quantities are just illusion. All what is there are just qualities and relational order (I know that all matters are not so simple like that, but is it true).

   Thus if the result from ±/+ is negative, then the result of the opposite ratio +/± must be positive, and so on with the other relationships. Hereby, we can deduce three relationships:
+x - = ± , + x ± = + , - x ± = -

   The relationship 1/+=- expresses about "inversing the relationship", it is the implicit statement about the relation between the three states, and its outcomes are the relative opposite relationships for both opposites with the neutral, and also for neutral with the two opposites.

   It remains to find the relationship between the positive and negative in the two directions, namely: -/+ and +/- where it is a direct relationship between the positive and negative without taking the neutral state into consideration. If we consider the negative relative to the positive is smaller, then: -/+=- but this is not right because it contradicts with the rest of the rules, whereas we get the sign rule: -x+=- which contradicts with –x+=±, consequently we have to reverse our assumption to get:-/+=+. By the same manner we get:+/-=-, thereby the reversibility of relations has been achieved in the two directions without any conflict with the other rules. But why here the same logic does not apply? Is that because of neglecting the neutral state, or for any other reason?

   One way to deduce +/- and -/+, and keeping all the meanings be consistent with each other; is by considering the states to be in a circular arrangement not linear, thereby +/- locates positive to be to the right of negative and we get + for that, and vise versa with +/- to get - (this is the essence of consistency where the relationships should be closed on itself, which is an essential requisite, in other words: if it is closed on itself, sure it is consistent; this this very another important principle).

   If we apply the same circular arrangement on the first set of rules, namely
(+, -) we will always get a contradiction, not only because it is already contains a contradiction; but because the right or left to any state will always meet the same state.

   In addition, the relative relationship between negative and positive can be deduced from (±/+x-/±=-/+=-x-) namely from the relative relationship between neutral and positive, then the relationship between negative and neutral in sequence to get the relation between the negative and positive (-/+) which equals –x- (because ±/+=-, -/±=-) thereby: -/+=-x-=+ (sure this not a deduction for the rule). Likewise +/-=+x+=-, and since the relation between -/+ has to be opposite to the relation of +/-, thus the result of –x- is opposite to the result of +x+. Thereby, the direct relation between positive and negative when overlooking neutral state results in the reversal of the relationship!!! But the principle of opposition of states still valid, the relation and the opposite relation are inevitably opposite.

   The Question arises: is he sign rules applicable on any category or any other logic taking into consideration that its deduction is not based on any category? My answer is: the logic of sign rules is based upon the cause and effect logic and tangible matters (if this is not clear up till this point here, it will be explained later in detail in the second part of the theory). Therefore, I assure its applicability on any physical related concepts.

   If we are talking about other forms of logic: for example if we talked about the formal relationship between even and odd concepts, it may refuse to get beneath any logic of sign rules, that is because it is related to abstract numerical logic that has no relevance to physical nature where: even x even= even, odd x even = even or odd, odd x odd= odd. Also, we can say that the friend of my friend is my friend (+x+=+), the enemy of my enemy is my friend (-x-=+), the friend of my enemy is my enemy (+x-=-), and  enemy of my friend is my enemy (-x+=-)  which all are ridicules despite its correspondence with the first set of sign rules, it doesn't represent actual facts.

   I already have written many chapters relating to different sciences to show the ability of the concept of opposites to unify all the human knowledge. Some of these chapters are: The deduction of the Euclidean geometry - Psychology and opposites - The axes system according to higher sign rules, and many much more and I will not publish them in the second part, because it needs more investigation, but it contains the basis to show how this all can be done (in the second part there is a chapter about differentiation and integration, can you imagine how is that related to opposites and the sign rules?).

   The truth is certainly simple, axiomatic, and intuitive – if it is not like that, then we have not apprehended the true relationships between things- all what we need is more analysis to realize the truth in its simplest form. If we actually could attain the simplest form, there will be no proof but all matters are absolutely obvious. What has been done here so far is not a proof, but it is likely close to self-evident facts, where the real relations between things are lost, and the complexity of inventions deceived us to the degree that we thought nature is complex, but it is the simplicity itself.

The Causal Principle within Sign Rules

   Both of the two opposites is a cause for the other where they are interrelated, and neutral is the cause for both of them.

   The deduction of both the first and second sets is based upon avoiding contradiction in the form of division, in spite of that the contradiction can appear in the form of multiplication without converting to division. If a state affects two different states, and the result is the same, this is a contradiction and a breach of causal logic – thereby as if the second state (the affected one) has no effect – no one of the interacting states can be without effect (except the neutral state), consequently this logic can be used to deduce the first and second sets without the transformation to the division form, which is called the causal principle.

   If the result of two interacting states does not change with the change of one of them, then the neutral state has a relation with that; since neutral state is not affecting, and to clarify that through the first set:

Suppose that +x+=+ namely a positive state affecting another one – if the same positive state affected negative state, then we must get a different result, thus: +x-=- and if this rule means the effect of negative state on another positive state, then if negative state affects another negative state, then we must get a different result: -x-=+ hence we get the first set of sign rules using this simple sequential logic by applying a very simple qualitative principle.

   By following the same reasoning with the second set: we find the effect of a positive state on another negative state results in neutral (+ x - = ±), whereas its effect on a positive state results in negative state (+ x + = -), whereas its effect on neutral state results in positive state (+ x ±=+). Accordingly the result differs with each affected state for the same effect, which is completely very logical and predictable, and the same logic applies for any other set of sign rules of higher degree (it is causal logic).

   By developing a computer program, we can easily deduce any consistent set of sign rules for any number of signs by applying this principle, and this goes as follows:
- Suppose any number of signs and suppose any symbols for them.
- Select the first sign and write all the possible rules with all the other signs taking into consideration to set a different result for each one (according to the causal principle).
- Select the next sign and write all the possible rules with all the remaining signs.
- Repeat the previous step until all the signs are exhausted (recursive procedure).

   However, this principle is necessary but not sufficient to get a consistent set, I will explain why and how to get a consistent set. So when trying to apply this principle using a computer program, you will get additional non-consistent solutions, because in fact, there is no existence for what is called the sufficient reason or cause. Moreover, this is not the only logic to computerize the deduction of a consistent set of sign rules.

   The ideological consequences of sign rules are impressive, because do really nature follow these rules to attain the optimum consistent total state, that is for adaptation and natural selection to occur? It is all stands beside the naturalism and Darwinian attitudes as will be discussed in detail.

   Now the causal logic is governed by evident and proved rules, not just narrative or rhetorical sentences waiting to be tested; hereby the theoretical basis is the sound ground, whatever you tell me about experimentation. As I said that in spite of the simplicity of these rules, but they rearrange the human knowledge and make it all unified under the same concepts.

   If you see this principle as self-evident, but do you have a proof for it? I say that the sign rules are the proof, because their consistency is sufficient to confirm all the meanings that it can express about whatever it is.

   The same logic applies on the multiplication table, and then on our whole scientific knowledge, in other words how the multiplication table originates as a series of consistent relationships? It is simply governed by the causal principle, thereby the effect of a certain number on any other one (or the interaction between them) should always be different, also the same result can be generated from different inputs (3x4=3x2x2=6x2=12) (+x+=-x± =-), which is self-evident and does not need explanation.

   Nevertheless, how the multiplication table is endless, or how the sign rules are closed onto itself? The answer is simply that the number of states of the multiplication table are infinite, whereas with sign rules we start with a definite of states.

   Later on, I'll explain how all the known scientific methods are governed by these rules, where all the meanings are inherent there and waiting to be dismantled. You will not believe that starting from Francis Beckon methodology passing by other methodologies till the recent modern ones; are all incubated here within the sign rules.

About the Meaning of Consistency

   The meaning of consistency is changing as science gains more advances in new disciplines. When non-Euclidean geometries were discovered, consistency got new dimensions. Consistency in its usual meaning is related to axioms; and how not only these axioms must not contradict each other, but also they must not give rise to theorems which contradict each other. Herein, in this theory, the meaning of consistency sheds the light on a new viewpoint on consistency, which is more stronger, well proved, and opens new paths for discussions about the whole concept of consistency.

   The meaning of consistency mentioned here has no relation with the known meanings of consistency, because the substitution procedure was never a measure for that, except after discovering sign rules other than the known ones. Here are no postulates, but looking for a closed consistent set of symbols, whatever its meaning is. But its meaning fundamentally is qualitative.

   An important issue, which is the amenability of the sign rules to substitute in each other, which is a sufficient reason for consistency and non-contradiction, and this, is by far and in itself; is the strongest evidence in comparison to any other logical structure of postulates.

   If we are talking about the different geometries, we may say that from primitive postulates taken to construct and build a new shape for the same postulates, hereby there is no search for consistency, and it is all a matter of getting the results from certain introductory assumptions, not investigating its consistency

   That way the sign rules dismantle all the probable forms about the structures in nature, where the nature of consistency on non-consistency here is in its utmost levels of clarity and certainty. The sign rules don't depend upon postulates, but it is inevitable logical structures. The geometrical theorems don't affirm the validity of their postulates at all.

   It is the strongest meaning of consistency ever known, it is self-evident, because the group of relationships is closed onto itself to satisfy each other, and you cannot imagine much more simpler and intrinsic concept other than that.

   So, what is known about consistency it is a curious situation, for there is no test, no criterion, of consistency. We have not noticed so far any inconsistency in ordinary Euclidian geometry, for an instance, but the future may bring forth what we do not know, and the entire structure may someday be seen to be consistent only under conditions not yet stated, which are really all come now to be true with the concepts of opposites, but the question poses itself: can all forms of disciplines, thought, logic.., accrue to a closed set of relations?

   Later on, a detailed discussion about consistency is presented when deducing the sign rules using a computer program.

Principle of Opposition of Relative Relationships

   The general logic entails considering the relative relationships are inevitably opposite in both directions, other than that makes the relationships of all things are the same.

   Suppose the relation of s1 with s2 is the same of s2 with s1, this means that: s1/s2= s2/s1then s1= s2, which means the neutrality between the two opposite states, namely the absence of any opposition between them and they have the same state. The relationship (physically) means the mutual effect, so if the effect is the same interchangeably, then it entails their equality. So in general the relationship of s1 with s2 is not the same of s2 with s1, namely the effect of each one on the other is different or opposite according to the value of the affecting and effected states.

   Thus, the concept of relationship shows the importance of division to express about the relation and its opposite, and the previous relationship can be rewritten in the form:

s1 x 1/s2 = s2 x 1/s1, this form lets both of s1, s2 to appear in opposite form in both sides (the value and its inverse, thereby to cancel each other) knowing that the value and its inverse are in relation to "one", in other words a relationship exists between state and its opposite through neutral state which is represented by "one", thus s1/1 is reverse to 1/s1, which we  all already know, but may not be proved qualitatively or we don't know its deep essence and implications.

   From the second set of rules: +/-=-, -/+=+ thus the relation of s1 with s2 is the opposite to the relation of s2 with s1, whereas with the first set: +/-=- , -/+=- and this is from the viewpoint of qualities is unacceptable and inconsistent, since the relation in the two directions can't be the same except in the case of their equality, and this what we actually get, namely: s1/s2=s2/s1 which corresponds to +/- = -/+, then (-)2=(+)2 , by taking the square root we get -=+ namely s1= s2. The reason behind getting this contradiction with the first set is that the ratio and its inverse are not opposite, so in the traditional mathematics and all branches of science we get for the square root of + two outcomes + or – which is considered acceptable and ok, but it is not, except with justification which will be tackled in many subjects. (Look forward under the subject "square and cubic roots" for more explanation, as will be seen: the absence of neutral state causes contradictions).

   Concerning the remaining rules of  the second set, we find the relation of positive relative to neutral "is opposite to" the relation of neutral relative to positive, namely: +/±=+, ±/+=- , also the relation of negative relative to neutral "is opposite to" neutral relative to negative namely: -/±=-, ±/-=+, thus all relative relationships in the second set are opposite.

   It remains the relations of a state with another similar one, which are unidirectional because "there is no relation and the opposite one". Therefore they all are relative neutral relationships that express about qualitative equivalence between the two states, where the relativity of a state to another similar one means neutrality, so we get: +/+=-/-=±/±= ±. Thereby the sign rules satisfies a general logical meanings concerning tangible reality and the logic in general, and what refuses to go in parallel with it is not logical or does not relate to physical reality, and it is just an invention or convention.

   The relation is represented by the ratio, because if s1>s2, then the relation of s1 with s2 is opposite to the relation of s2 with s1, and to represent that we can say that s1/s2>1, s2/s1<1. So, the relationship relative to "one" is a standard reference, or to say that: s1-s2>1, s2-s1<1, and with subtraction the ordered relation differs horizontally for the two terms, but with division it differs vertically. It is just symbolization and no meaning related to division or subtraction, our brains have been biased, suffering severe deficiency and misleading concepts.

   Instead of saying that: +/-=- it can be rewritten in the form (+,-)=- , thereby: -/+=+ corresponds to (-,+)=- so we get ordered pairs, and it would be: +/+=-/-=±/±=± which corresponds to (+,+)=(-,-)=(±,±)=± respectively, and -/±=- corresponds to (-,±)=-, and +/±=+ corresponds to (+,±)=+.

   Namely, the two pairs (-,+)=(-,±)=- are equivalent, where their relative relationship is the same. So, we can say that symbolic representation used with the sign rules (also the entire science as will be explained later) does not mean division or multiplication, it is mere a symbolic method to express about ordered relations.

   The principle of opposition of relative relations is one of the most basic principles for all the forms of entities – is it essential and preliminary or derived from other ones? This principle is a different formulation of Newton's third law that states: "For every action there is an equal and opposite reaction", it is almost the same content with different words. If Newton expressed about it using action, he is right, since it is a general formalization that can be applied on all subjects. The principle of opposite relative relations is a general logical formalization, it decides that relative relations are equal and opposite, in causal language it means that the equality of result with the cause, which is imperative for conservation laws, otherwise there would be loss or creation from void.

   Where that principle was in the human thought, and what is its importance and value? It is a basic principle of that real logic not formal logic, it rules the nature.

   The manifestation of absence of this principle from the first set of rules is the collapse of the meanings of increase and decrease and the contradiction for both positive and negative states. The other manifestation is that the square root of positive state has two outcomes: either positive or negative, namely all expectations are possible, namely it does not determine any thing which is tautology.

   If the relation of s1 with s2 is in the form: s1=2s2 then the corresponding opposite relation would be: s2=1/2 s1, then we get: s1/s2=2, s2/s1=1/2, then we get: s1/s2 x s2/s1= 2 x 1/2 = 1

But if the relation was not reversible in the two directions we get:

s1=2s2, s2=2s1, namely the relation is the same in the two directions, and hence s1/s2=2, s2/s1=2, thereby this is an assertion that their relation is not opposite, where it is supposed that one of them equal 2 and the other 1/2 to cancels each other, and such a relationship when substituting from one into another results in: s2=1 (this relation is the same like that we got before with table reversal, and considering actions are neutral) which is satisfied when s is equal to one, namely considering the states are neutral in both directions, so this formula is satisfied with neutral states.

   The simplest relation that is probable to inversion, namely the possibility to do mutual equivalent operations in the two directions that is to say: a=b , b=a which results in 1=1 (this is exactly the neutrality and they are the same thing, the mutual relation is not opposite and hence the states are neutral). Then, comes after it in complexity: a=1/2b , b=1/2 a, which means that the relation of a, b is reversible, it is the same in both directions and not opposite (instead of a=1b , b=1a, the one is replaced by 1/2 or may be replaced by 2 and we get the same result) thus we get: 2a=b, 2b=a , by substitution from the first in the second we get: 2 x 2a=a , 4=1 which corresponds to s2=1 which is satisfied when s=1, namely by considering neutrality of states to satisfy the relations in the two directions, so it is satisfied with neutral states.

   For each action; there is an equal reaction in magnitude and different in quality. The principle of opposite relative relations is a logical imperative principle rules all entities, it simply means the unbiased of one state over the other, namely they are both equivalent; accordingly, this principle satisfies the symmetry and systematic relationships.

   To sum up: state relative to itself is absolutely neutral, so that: s/s=1 or (s, s)=1 or any other possible form which is maintained in a systematic digression. Also the relationships of opposites is opposite in the two directions. With states x, y the relation of x relative to y is reverse to the relation of y with x, thereby x/y * y/x=1. Also the contradiction present in the first set: (+)2=(-)2 , then +=- is a result of the non-reversibility of relationships between the opposite states (or they are not opposites and a neutral states is lurking there as we will see). At last, not least the symbolism used with sign rules does not mean multiplication or division; it is only a representation to set up relationships.

Importance of the Neutral State

   Suppose two points A, B (with the same visual order), where does point A lie? It lies to the left of B, and where does point B lie? It lies to the right of A, i.e. the relative relation is opposite in the two directions. If we want to express about that symbolically, it can be: A/B=A, B/A=B, thus the product A x B is susceptible to two answers A and B (since the two points A, B are supposedly to be in opposite locations, but the result of their relationship comes out to be neutral which is contrary to the first supposition).

  As such, the importance of neutral state imposes itself to solve the contradiction. The removal of neutral state results in the equality of the relations in the two opposite directions.

   Is our supposition that A being to the left of B without a separating state between them, is an erroneous and subjective judgment, and to be objective there ought to be a third point in between? It is actually subjective because we intruded ourselves in the judgment unintentionally, as if we are an external point observing the direction right or left, otherwise we could not determine that direction, so we are the reference point.

   The motion between two points in different directions is absolutely equivalent and we cannot realize the direction of motion except in the presence of a third reference point. When moving between two points, there is no way to know if we are moving from A to B or the reverse direction. The only method to discriminate that is the presence of a third reference point, where from its relative position we can determine the direction. Supposedly, If the reference point is to right, then we are moving from A to B and vise versa. So, if we decided that A is to the left of B, and B to the right of A, there must be an external global view of ourselves observing the direction of motion and keeping that in our memory, then comparing it with the motion in the other direction.

   Our direction can't be identified except relative to a third reference point, that settles opposition between states, so can you know if you are moving to Alex or Cairo except through the guiding symbols or a specific identifier you know on any of the two sides?! There is impossibility to know the direction of motion without third reference point, and from its position relative to motion direction, where that direction determines whether it is from Alex to Cairo, or from Cairo to Alex.

   The state is being absolutely neutral if not referenced to any other one. If we supposed that x=y/z where y≠z, that is y can't be equal to z, i.e. x consists of a ratio (namely state x is determined by reference to other two states, so we have three states). If the ratio disappeared we must get x=1, in that way reaching neutrality. Thus if it becomes y=z this means that they are the same, and consequently the meaning of ratio collapses and thus z/y=y/z where both of them is the same, consequently any relationship is essentially has three states not two, and if it is dual the opposition disappears and one of the states transmuted to the other.

   The positive (surplus) and negative (dearth) has been defined only because they get aside from neutral (equality). Excess or shortage is just a ratio relative to equality position- increase or decrease does not come from outside but by decomposing neutral state, then increase occurs in one section and decrease in another, so if neutral is 22 it becomes 31 or 13 thus increase or decrease are relative to initial position 22.

   The essence of state is absolutely devoid of qualitative meanings– and to consider state to have a quantity, this negates its being as absolute state – i.e. not to be absolute positive or absolute negative – this means positive state must contain negative state and vise versa.

   In other words the positive state can contain an amount of neutral which is responsible for qualitative side – where neutral at first glance appears to express mainly about state, but it is actually what creates quantity of state, so neutral is the link between quantity and quality, because it is characterized from qualitative side to be as state, and characterized from quantity side by stability or un-changeability.

    How do we say that the basis of everything is qualitative, and then we talk about the distribution of quality over quantity, by that: is quantity is another kind of quality? It is actually so since it is just a neutral state, so it has no effect and we thought it is a quantity. The fact that the basic concept is the qualitative not the quantitative is so clear and simple to such degree that it is undisputable or debatable, we were just in need to show how quantity is in opposition with quality – and this is clear from the fact that quantity is represented by neutral state in opposition to quality represented by (+,-).

Is There a Contradiction in the First Set of Sign Rules?

   With the first set of sign rules: +x+x+x+…..=+, any number of positive states has no effect because it does not change state, whereas decrease accrues to reversal of state (-x-=+, -x+=-) where the second state has changed and this is a doubtful logic. The right logic for qualities is different from what is known. As if the positive state in all scientific equations represents steadiness and non-changeability; whereas "negation is the action" which is a well known mystic quotation, which had been cited in many philosophies and modern writings till today.

   The positive state arises from the first set as a ratio between similar states: +/+=+, -/-=+ if this signifies something, it signifies its neutrality, but negative state originated from the ratio of different states: +/-=-, -/+=- thus the ratio and its reverse conduces to the same result (which is the utmost of nonsense, so how a relation and its opposite conduces to the same state except they are equivalent) which accrues to: +=- because it is generated from taking the square root of the two sides: (+)2=(-)2 and what asserts that (if we are not pleased with deducing +=- by taking the square root) is that with the first set the square root of positive state is positive or negative, which assures the equality of these outcomes, so if the result of an operation is probable to two outcomes implies a contradiction, or it implies that they are equal? In other words it does not purport useful answer and does not give decisive decision, and we will show the right interpretation of the first set later on. The contradiction is not necessarily to be so - because from another viewpoint it can't be like that.

   What again assures this contradiction from that: +/-=-/+, thus: +/- x +/- =1 , (+)2 = (-)2=1 = +, whereby "one" signifies neutrality, so does any one believe that +/- is the reverse of +/- and differs from it? (There are no mistakes in typing) except their result is neutral, and this is the compulsory meaning of the positive state. In other words since it has been evident that the positive state results from division of similar states, consequently: (+)2=(-)2=(+/-)2 by taking the square root of the two sides we get: +/-=+, thereby: +x-=+ which is contradictory to the set of sign rules,  or its result should to be: -=+ which is a mandatory result without any evading outlet or escape from such bitter facts.


Distribution Rule and the Sign Rules

   It is known that the sign rule: -x-=+ results from the distribution rule which means: 2(x+y)=2x+2y. This sign rule results as follows:

(-2) (-3) = (-2) (-3) + (0) (3)

              = (-2) (-3) + (-2+2) (3)

              = (-2) (-3) + (-2) (3) + (2) (3)

              = (-2) (-3+3) + (2) (3)

              =  (-2) (0) + (2) (3)

              = +6

   This is how –x- is converted to + without applying any sign rules in the steps, just the distribution rule. Is that proves that –x-=+ and it is unique and not probable to give another answer?! But what makes us sure that it follows the first set of rules not the second set, because this rule is common between both of them? Also the first set is not justified, because it has a reverse form which stand equally with the first set, in which –x-=- not +. Is it a result from the second set, because it has only one form? Here no sign rules have been applied except in the final step and considering the result is (2)(3)=+6, i.e. applying the rule +x+=+, and this has no justification or assertion that relates to the first set, because it can be considered corresponds to ± x ± = ±, because the positive sign has been put compulsory (who believes that the whole edifice of science is based  on un-understood sign rules?). For your knowledge the axes systems needed 300 years to settle down, which is based mainly on the sign rules of the first set. This long period makes anyone in doubt about the clarity of the beginnings of the basic concepts.

   But what is the relation between the distribution rule and the concept of distribution itself when deducing –x-=+, and why it is compulsory? It seems to be the modal form for any logic, and it seems at the beginning of inventing algebra, this rule of distribution has been discovered – and around it other matters has been established (as I said the known axes system needed 300 years to settle down, and you can imagine what happened in this long period of trial and error).

Relativity of States (General-Special-Unique)
   Dealing with three concepts: General (g), special (s), and unique (u). For example plant is the general, grain is the special, and wheat is the unique. Grain as special is considered general to wheat, and is considered unique to plant, from that we can deduce:
                      = s/u=±/-=+=g  , u x g = s , + x - = ±
                      = s/g=±/+=-=u , u x g = s , + x - = ±

   It is evident that special which is the middle term has neutral state, whereas the extremes are opposites, this is by taking into consideration two terms and overlooking the third term. The original trio situation considering the middle term can't be defined except in the presence of the other two states – relatively to both – we can then express about that in the form:
Grain/(plant x wheat)= special/ (general x unique) = special
Special x special = general x unique , ± x ± = + x - , and its detailed form:
(grain/plant) x (grain/wheat)= grain2 / (plant x wheat) = special
Namely: s2 / g x u = s , s= g x u , ± = + x –

   The relation of special with general (which is unique) is equivalent the relation of unique to special (which is unique too), namely: special/general=unique/special, special x special = general x unique, whereas unique to special is considered unique, namely: u/s=u, u x s = u, - x ± = -.

General (g) to special (s) is expressed as: g/s=+/±= +, unique (u) to special (s) is unique (u): u/s=-/± = - = u
Whereas to find the relation of unique and general where they exist in two non- consecutive levels, this can be done through the intermediate term which is special in that way: unique/special x special/general = unique/general
u x u = u/g , - x - = -/+, +=+
(The relation here goes from unique to general, and the reverse relation can be formulated from general descending down to unique thereby:
g/s x s/u=g x g = g/u , + x + = +/- , -=- , going in the reverse direction results in the reverse states).

   It is clear that the two non-consecutive relations is equal to the relative product for the consecutive terms, supported by the evidence of the result: unique/general from the two relative relations, also it is evident that the result of unique and general relation is u x u, which must accrue to g, namely the general term (this is according to the sign rules), thereby unique to general is general, how does this occur? Do they have the same state? Or does the relations reverse and exchange places since unique to general becomes general (general to unique is unique, so positive to negative is negative)? Or is there a circulation in relations, so it is not a linear but circular relation? Or are there a root and branches, but branches can be roots? Or do we make the result u x u without conducing to a single term? Or are the relations inevitably exchangeable and do not only going in one way; and other than that is absolutely wrong?

   The meanings related to deducing from general, special, or unique are irrelevant, it is just an expression about hierarchical relational levels, and those relations are just a segment has been cut from more generalized concepts, with the possibility of transforming positive into negative or vise versa.

   Unique is not a direct function of general, but this is done through special, if both positive and negative is defined without taking neutral into consideration, this accrues to the reversal of states, and the moment that neutral disappears, is the moment when states are exchanged.

   General (g) relative to unique (u) is unique, but is that a result of isolating the middle term, which is an essential term in setting up the relation between them? Is it initially possible to define positive and negative without neutral? Or is its presence mandatory for the definition of opposites? Both opposites are known by their dissimilarity and in contrast there is the equality, thereby there must be three states together, consequently trying to isolate neutral reverses the states and positive becomes negative and negative become positive.

   For unique to be transformed into general, it should be divided into special and unique, namely: - = ± x - = + x – x - = + x +, thus the negative transforms into two positive states, thereby it can get into relation and unique becomes a new general (not necessarily the same original general). Each unique (-) can become general (+) by dividing it into special (±) and unique (-).

   Negative relative to positive, if neutral in between has been removed is considered positive, and vise versa, they become the same state which is the reference state – is that from its viewpoint? Namely negative gets the same state of positive when referenced to it – whereas removing neutral means the two states become the same one that is referred to. So that general (plant) to unique (wheat) if the special (crop) is removed becomes unique, namely general transforms into unique.
Transformation between the two opposites
   For any state to be transformed to its opposite, it must first pass through neutral state, because the opposite lies on the other side- thus neutrality takes place before transformation to the other opposite. If we have state a positive state +, it must be converted to ± then -, and this occurs with the consecutive effect of two negative states, thereby: +x-x-=±x-=-, namely in the first step with the effect of negative state on positive: -x+=± to get a neutral state, then with a second effect of another negative state we get negative state:-x±=-.

   For instance, to exchange states between two bodies where one of them is hot and the other is cold; and for the hot body to become cold and vise versa, they must go through a moment where the temperature of them is equal then the states can go in reverse direction. The natural tendency between them is the thermal equilibrium, and for either state to be reversed a work must be done. The cold body (-) to become hot (+), this occurs only when exerting a work by a positive state: to get +x- = ±, and then another additional positive effect: +x±=+. To the contrary, for the hot to become colder, this is done by a negative factor or more negativity: (-x+)x-=± x - = -. So, there no direct connotation that +x+=- or that –x-=+, but a change takes place when two opposites exist in the presence of a third affecting state, which is higher or lower than any of them (in fact as if there are actually three different states +,-,± if we have cold and hot and much hotter, or cold and hot and much colder, so there are two extremes and one intermediate state).

   In other words we can say that: for a state to be converted into its opposite, it must be decomposed into two opposite states: if we have the state (+) it can't be transformed solely, but there must be another state (+) that is decomposed into –x-, then +x- combined to get ±, then from the interaction of ±x-, a negative state – is produced, thus it is to be: +x+=-. Ergo, this relation can't be understood except in this sequence, namely there must be decomposition for one of the states, and this is an implicit result that: –x-=+ but not a deduction for it. So, this interprets how +x+=- and –x-=+ (using the relationships: + x - = ±, + x ± = +, - x ± = -).

   Therefore we have to understand +x+ to be +x-x-, because positive state neutralizes with negative, and negative remains affecting on the neutral state which accrues to emerging of a negative state. If we have hot body (+) and another cold (-), then the natural tendency is the neutrality and remains the resulting state to determine the nature of the affecting state.

The Mathematical Meaning of Neutral State
   The traditional mathematical concepts impose the conventions that gain is positive while loss is negative, but taking neutral state into consideration makes it represents the intermediate situation between gain and loss, which is "possession or ownership". Traditional mathematics without neutral state considers that we never possess anything, but we are continually gaining or losing all the time, and we add to our losses or to our gains, not to what we already have. This situation is far from reasonability that is attained with the existence of the neutral state, where its absence makes matters indiscernible and more obscure.

   Suppose a person owns 100 pounds (±100), afterwards he losses 50 pounds (-50), then he has to have 50 pounds (±50), thus: ±100-50=±50. Then suppose a person owns 100 pounds (±100), afterwards losses 200 pounds (-200), then he must be debited by 100 pounds (-100), thus: ±100-200=-100. Thereby, the sign rules that we can conclude from the relationship of possession with loss are:
± (+) - = ± if ±>- , ± (+) - = - if ->±, ± (+)-=0 if ±=-.

   Suppose a person possesses 100 pounds (±100), afterwards he gains 50 pounds (+50), thus: ±100+50=±150, this means in the new situation he must possess 150 pounds, and we can't say he earned 150 (+150) because this not what actually happened. Hence the relationship between possession and gain is: ± (+) +=±, regardless of the value of ±, +.

   The interpretation of these relational sign rules using possession with both gain and loss, would be clear when transferring from the second set to the first set, which simply is done by replacing the neutral state with positive state, and consequently the previous deduced relations become:
+ (+) -=+ if +>-, + (+)-=- if ->+, + (+)-=0 if +=-, whereas: + (+) +=+, regardless of the values of both +, +.

   It is evident that we have got the traditional concepts used in mathematics; thereby possession makes matters more detailed and specific by differentiating between possession and gain.

   The transformation of neutral in the second set to a positive state in the first set can be interpreted as if the first set unified positive and neutral in one state and considered them a gain, and this is to some extent is acceptable, moreover … the act of the neutral state in the second set is like the positive state in the first set, since both of them does not change state with multiplication.

   With multiplication the relation of neutral with gain or loss has no meaning, because there will not be a problem stating that: a person owns 100 pounds (±100), then find what he owns (or gains) after 3 months (+3)? The question is meaningless or tautological with possession (since he should still owns the same amount) because it is timeless process, namely it is fixed and not changing in time, since this is the nature of the neutral state.

   But simply it can be formulated another way: three boxes each contain 100 pieces, how many pieces are there? No action and no time. So, with multiplication the repeating operations has meaning only in the absence of time concept, since the neutral state is steady, namely neutral is timeless state representing constancy, thus with multiplication it still has a meaning.

Square and Cubic Roots
   Executing square and cubic roots on numerals is different from doing it on states, whereas the operational order is effective with states that results in different outcomes, but with numbers it is indifferent.

   From the first set of rules: +x+=-x-=+, thereby (+)2 = (-)2 and taking the square root can be done in two ways: first, is to take the square root then squaring: (√+)2= (√-)2, and in that case we cannot find out the square root of negative. The second procedure is to do squaring first then taking the square root: √(+)2=√(-)2 then we get √+=√+, +=+ or -=-, hence what should be is doing squaring first then taking the square root, because the starting formula (+)2 = (-)2 includes squaring then applying the square root on both sides, and the starting point is not the opposite, not because of the impossibility of finding the root of negative.

   By applying the cubic roots according to the first set; we find that the cubic root of the positive state is positive, and the cubic root of the negative state is negative, that is because the product of three similar states results in the same state whether it is positive or negative (+x+x+=+, -x-x-=-).

   Now comes the turn to apply the same operations according to the second set, either with square or cubic roots, which can be done in the same manner. With square root we find that: we know that +x+=-, -x-=+ then with (+)2= -, first by applying on both sides the square root then squaring: (√+)2=√- then we get (-)2=+, +=+. But squaring then applying square root: √(+)2=√-, we get √-=√-, then +=+ the result is the same regardless of the order of operations, and this not a general judge as will be shown.

   Finding the square root can be done on different forms of rules:
+x(±)=-x- both sides have positive result, +x (+x-)=-x-, (+)2 x - = (-)2 by applying the square root we get: +x √-=-, +x+=-, hence the square root results in reversing the original states, instead of being positive in both sides it became negative. Also, since +x+=- by applying the square root we get: √(+)2=√-, +=+, hence taking the square root reverses the states. Using other forms like: (+)3=± and by rewriting it in the form: (+)2 x +=±, by applying the square root we get +x-=± which is all right and valid.

   With applying the cubic root we find that: +x+x+=-x-x- hence: (+)3=(-)3 by applying the cubic root we confront two alternatives: performing cubic root on state then cubing, namely it is written: (3√+)3=(3√-)3 but it is undetermined for both sides, because there is no three similar states that have a product to be positive or negative. The second alternative is done by applying cubing then the cubic root, which is written: 3√(+)3=3√(-)3 hence the result of the two sides is neutral, because what is under the root sign results in neutral state in both sides, and the cubic roots of the neutral state results in a neutral state. But we repeat again that the most acceptable is applying cubic root after cubing because this is the origin of the equality, thus applying the operation on both sides becomes possible.

   Since with the first set we get two outcomes for the square root of + which are + and – since +x+=-x-=+, and the same applies to the second set but with cubic root, so that the cubic root of ± can be + or – since +x+x+=-x-x-=±, and this all have many important implications and consequences about our mathematical knowledge and the knowledge in general:
Now it is clear and asserted that using the first set is unjustified at all, and it is not the only form of sings that can be used, because what we thought as impossible with the first set reappeared with the second set but with cubic operation (I mean square and cubic roots), so the second set can replace the first equally, and for far and beyond objectives or purposes, because it is sure more general and includes the first.

   Also we realize the equivalence between the positive state in the first set and the neutral state in the second set, since both of them is ineffective. Later on, other sets of rules will be deduced of the third and forth degree, for four and five states respectively; and their meanings will be explained, and all sign rules are deduced using the same logic, which signifies the unity of the whole universe that is controlled by the same logic on all levels of complexity.

   Making a comparison between the numeric values and states, for example when we set: (4)2= (4)2 or (4)3= (4)3 we can easily guess that we would get 4=4, but with (+)2=(-)2 can we guess directly that +=-?

   The meanings are different, and without any hesitation we can establish one to one relationship with numbers, because of the nature of numbers and considering them as neutrals that is different from states.

   We can't say that the square and square root cancels each other directly as operations, but they have to be applied on states. So, with the first set when setting (+)2=(-)2 and then applying the square root, sure we will not get +=-, but we have to apply the inner operations then the outer, so we get +=+, and the same logic applies with (+)3=(-)3 according the second set.

   Many questions that may arise: what is the treatment of such operations with higher degrees of sign rules? Is it acceptable for an operation to have two outcomes where the square root of positive is either positive or negative with the first set? Is there in general in nature a meaning for an operation that is probable to two results and to accept one and refuse the other? Can the cubic root be impossible with the second set, because it has no physical meaning, because there is duality in nature? So there will be always one definite outcome.

   To sum up: with the first set the square root of the positive state results in + or –, whereas the square root of negative state it is impossible; whereas the cubic root results in + for + and – for -. With the second set the square root is possible for any state, and cubic root is impossible for positive or negative, but possible for neutral. With cubic root of ± we can get two outcomes + or -.

Not finished yet (Under translation)

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