# Introduction and Foundation Concepts for Itoequations

#### Introduction

The purpose of itoequations is the representing the ideas of awaretheory in symbolic and mathematical forms. Frequently, new ideas and concepts can be formed symbolically simply by manipulating the symbols following syntactic valid rules. when the equation are formed correctly they can be more precise in meaning than the verbal equivalent.

These following equations will be represented in two ways. The first line is a name and a simplified version of the equation. We call this first part the name or number of the equation. The second line or lines will be an elaboration of the equation. We call the second part the elaboration.

The reduced equation (the shortest form of the equation usually in the first line) will give information about the equation in a more simplified or shorted form. The first number before the period distinguishes the equation grouping. The second number indicated what the equation deals with in terms of originals, idoriginals, cidentireplicas etc. The third symbol or letter deals with the relational operator. The forth symbol or letter deals with the functional operator. The fifth section defines the variables in parenthesis that vary. Between the different sections can be periods.

The simplest reduction of an equation is a number as in equation “1”. It supplies little information. The next simplest reduction is a name like “citomultiplicity”. When equations relate to verbally defined concepts that have names, the equations will often be labeled with these names. The third level of reduction or elaboration is the simplest symbolic equation or SSE. The SSE through explicit rules allows for the elaboration of level four. The fourth level is where there is actually an equating symbols relating two or more things. Equations can be composed of different levels as long as there a clearly defined way of reducing or elaborating the different levels. There is no defined limit to the amount of elaboration that these equations can be submitted to. Eventually an elaboration can include very specific information about a concept or object.

There is a numbers attached to itoequations for the specific purpose of designating which type of itobody is being related 0 is for itoidentireplicas, 1 for originals, 2 for idoriginals, 3 for coriginals, 4 for cidentireplicas, 5 for videntireplicas, 6 for fidentireplicas, 7 for isoidentireplicas, 8 for enhaidentireplicas, 9 for musidentireplicas, 10 for insidentireplicas, 11 for tridentireplicas, 12 for nrgidentireplicas, 13 for comboidentireplicas, 14 for simidentireplicas,. So a beginning designation of (1.4) on a itoequation would relate originals and cidentireplicas. "4.7" would relate cidentireplicas and isoidentireplicas.

A number of simple redundant equations will be used to give a rudimentary working knowledge for this mathematics of consciousness before we come to the equations that the identity theory generates as principles and properties of consciousness in this universe.

## Simple Expressions

1. $Ori(E)$ This term or expression is the matter related properties of an original.
2. $Cito(U)$ This term or expression is the universe that the cidentireplica is in. A different universe can have the same set of physical laws or any number of different physical laws than this universe has. It can also represent a different place and, or time in this universe where there are different physical laws as well that can effect the properties of the cidentireplica.
3. $Ori(C,X) \; or \; Ori(M)$ This term or expression means the consciousness and ixperiencitness or mentality of an original.
4. $Ori(O,U,E,D,T,S,F)$ This term or expression is a listing of physical properties of an an original.
5. $Ori(O,U,E,D,T,S,F,B,C,X)$ This term or expression is a listing of the physical and mental properties of an original.
6. $Fid(O,U,E,D,T,S,F,B,C,X)$ This term or expression is a listing of the physical and mental properties of a fidentireplica.
7. $Iso(S,F,M)$ This term or expression is the structure, functioning, and mentality(consciousness (C)and ixperiencitness (X)) of an isoidentireplica.

## Simple Oriequations

Oriequations are equations that relate originals to each other, or to other types of itobodies.

The symbol $\blacktriangleleft$ replaces for simplicity sake the expression $\xleftarrow{Compaction} \dot= \xrightarrow{Elaboration}$. Because of the complexity of consciousness and the structure and functioning of consciousness producing bodies The concept of compaction and elaboration of concepts is very important. One symbol can represent an extremely complex concept or complex group of interrelated knowledge. For example, the word Bob it is only three letters or a sound. Yet it can actually refer to a specific person which is a elaboration of identity of object to word. There can be a very large sequence of elaborations until every aspect of the original word is exactly defined. For instance, to the placement of every atom in bob's body at every instance in time. And to the consciousness that is produced by bobs body as every moment in time

The equation $Ori(C,X) \equiv Ori(M)$ is a replacement or compaction equation where one expression can be replaced by the other as needed. This equation means the that consciousness and ixperiencitness of the original is the mentality of the original. This is a definition type replacement equation not a scientific or empirical replacement equation that can be true or false determined by experimental evidence.

Definition based equations versus evidence based equations.

The statement that ixperiencitness actually exists is true or false based on the definition of ixperiencitness and then if it is an empirically based concept if it can be tested as to its actual existence. The concept of existence is not a simple concept because there can be a number of kinds of existence. For instance, there is current physical existence like a rock or other object. There is also effective existence where the concept is not an object but still effect people or other objects. For instance, the fictional character Tom sawyer, he does not exist as an object but has none the less effected people that have in turn changed reality that would not have happened other wise.

The number 1 represents the concept of the original (you). If the number has a prefix like this expression $1_u$ it means any original.

One of the simplest equations is:

$1.1\stackrel{=}{=} \; \;\xleftarrow{Compaction} \dot= \xrightarrow{Elaboration} \; \; Ori(O,U,E,D,T,S,F) == Ori(O,U,E,D,T,S,F)$

The compact equation ${1.1\stackrel{=}{=}}$ means that an originals is identical to itself when the two have the same identical original (O), Are in the same universe with the same universal physical laws (U), are made of the same matter/energy (E ), in the same dimensions/space (D), in the same time (T) , have identical structure (S),and functioning identically (F).

The equation $Ori(O,U,E,D,T,S,F) == Ori(O,U,E,D,T,S,F)$ is like saying 1 = 1. It is a tautology based on the definitions of 1 and =.

The compact equation ${1.1\stackrel{=}{=}}$ is a combination of the following equations:

1. $1.1F \; \; \blacktriangleleft \; \; Ori(O,U,E,D,T,S,F) =\!\!F\!\!= Ori(O,U,E,D,T,S,F)$ This equations represent the idea that an original will function exactly like itself. This is a consequence and sub case of equation 1.1.=. This is also true because the functioning in one term is the same as the functioning in the other term: F = F.
2. $1.1M \; \; \blacktriangleleft \; \; Ori(O,U,E,D,T,S,F) =\!\!M\!\!= Ori(O,U,E,D,T,S,F)$ This states that the mentality of the original is identical to the mentality of the original. If there is no mentality produced by the original the mentality produced is still the same, that of nothing.
3. $1.1\stackrel{=}{S} \; \; \;\blacktriangleleft \; \; Ori(O,U,E,D,T,S,F) =\! \!S\! \!= Ori(O,U,E,D,T,S,F)$ This states that the structure of the original is identical to the structure of the original (itself). If there is no mentality produced by the original the mentality produced is still the same, that of nothing.
4. $1.1\stackrel{=}{B} \; \; \blacktriangleleft \; \; Ori(O,U,E,D,T,S,F) =\!\!B\! \!= Ori(O,U,E,D,T,S,F)$ It is hard to argue with the idea that a person will behave exactly like itself when all factors are the same. This is a consequence of Equation 1.1.F because identical functioning produces identical behavior, and a sub case of equation 1.1=.
5. $1.1\stackrel{=}{T} \;\; \blacktriangleleft \; \; Ori(O,U,E,D,T,S,F) =\! \!T\! \!= Ori(O,U,E,D,T,S,F)$ By definition the time that the original exists in will be the same as the time it exists in.
6. $1.1\stackrel{=}{D} \; \; \blacktriangleleft \;\; Ori(O,U,E,D,T,S,F) =\! \!D\! \!= Ori(O,U,E,D,T,S,F)$ By definition the place that the original exists in will be the same as the place it exists in.
7. $1.1\stackrel{=}{U} \;\; \blacktriangleleft \;\; Ori(O,U,E,D,T,S,F) =\!\!U\!\!= Ori(O,U,E,D,T,S,F)$ By definition the universe that the original exists in will be the same as the universe it exists in.
8. $1.1\stackrel{=}{U} \;\; \blacktriangleleft \;\; Ori(O,U,E,D,T,S,F,C,X) =\!\!U\!\!= Ori(O,U,E,D,T,S,F,C,X)$

$1.1\stackrel{=}{U} \;\; \blacktriangleleft \;\; Ori(O,U,E,D,T,S,F,B,C,X) =\!\!U\!\!= Ori(O,U,E,D,T,S,F,B,C,X)$

Equation $1.1\stackrel{=}{=} \blacktriangleleft (1.1\stackrel{=}{M} \;\; \uplus\;\; 1.1\stackrel{=}{B} \;\; \uplus\;\; 1.1\stackrel{=}{P} \;\; \uplus\;\; 1.1\stackrel{=}{N} \;\;\uplus,\;\; 1.1\stackrel{=}{O} \;\; \uplus \;\;1.1\stackrel{=}{U} \;\;\uplus\;\; 1.1\stackrel{=}{E} \;\; \uplus\;\; 1.1\stackrel{=}{D} \;\; \uplus\;\; 1.1\stackrel{=}{T} \;\; \uplus \;\;11\stackrel{=}{S} \;\; \uplus\;\; 1.1\stackrel{=}{F} )$

This is just a statement of the definition of the relational operator ==.

Equations 1.1P, equation 1.1.N, equation 1.1.S etc. all follow from equation 1.1.= by equation 1.1.=…

$1.1\stackrel{=}{=} \;\; \blacktriangleleft \;\; Ori(O,U,E,D,T,S,C) == Ori(O,U,E,D,T,S,C)$

The symbol $\;\; \blacktriangleleft\;\;$ means that an elaboration will follow in the next equation.The process of elaboration is important because it allows more information to be represented in an equation.

$Ori(O,U,E,D,T,S,F) == Ori(O,U,E,D,T,S,F)\;\; \blacktriangleright \;\; 1.1\stackrel{=}{=}$

The symbol $\;\; \blacktriangleright\;\;$ means that a reduction of the previous equation will follow in the next equation. The process of reduction or simplication is important because it allows one to see the bigger picture so to speak and it takes less space to represent it this way, and it is easier to manipulate.

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## Simple Idoequations

Idoriginals are naturally occurring cidentireplicas of originals. Since they will occur naturally they are originals. And they are identical. They will be identified by the numeral 2 in the name of the equation.

$2.2\stackrel{=}{=} \;\; \xleftarrow{Compaction} \dot= \xrightarrow{Elaboration} \;\; Ido(O,U,E,D,T,S,F) == Ido(O,U,E,D,T,S,F)$

An idoriginal is identical to itself in all ways.

$2.2\stackrel{=}{M } \;\; \blacktriangleleft \;\; Ido(O,U,E,D,T,S,F) =\!\!M\!\!= Ido(O,U,E,D,T,S,F)$

This is a valid equation as well as $2.2\stackrel{=}{B}, 2.2\stackrel{=}{P}, 2.2\stackrel{=}{N}, 2.2\stackrel{=}{S}, 2.2\stackrel{=}{E},$ etc. because they are subcases of Equation $2.2\stackrel{=}{=}.$

$2.2\stackrel{=}{=} \;\; \blacktriangleleft \;\; (2.2\stackrel{=}{M} \;\; \uplus\;\; 2.2\stackrel{=}{B} \;\; \uplus\;\; 2.2\stackrel{=}{P} \;\; \uplus\;\; 2.2\stackrel{=}{N} \;\;\uplus,\;\; 2.2\stackrel{=}{O} \;\; \uplus \;\;2.2\stackrel{=}{U} \;\;\uplus\;\; 2.2\stackrel{=}{E} \;\; \uplus\;\; 2.2\stackrel{=}{D} \;\; \uplus\;\; 2.2\stackrel{=}{T} \;\; \uplus \;\;2.2\stackrel{=}{S} \;\; \uplus\;\; 2.2\stackrel{=}{F} )$

When we combine equation including originals and identical originals we use the combination of the numerals 1 representing originals and 2 representing idoriginals.

$1.2\stackrel{=}{=} \;\; \blacktriangleleft \;\; Ori(O,U,E,D,T,S,F) == Ido(O,U,E,D,T,S,F)$

This is true because of Equation 1.1= and PNI. All the subcases of this equation are valid as well. Such as $1.2\stackrel{=}{M}, 1.2\stackrel{=}{B}, 1.2D, 1.2\stackrel{=}{S}, 1.2\stackrel{=}{T}$, etc. because of equation $1.2\stackrel{=}{=}\supseteq$.

Equation $1.2\stackrel{=}{=} \;\; \blacktriangleleft \;\; (1.2\stackrel{=}{M} \;\; \uplus\;\; 1.2\stackrel{=}{B} \;\; \uplus\;\; 1.2\stackrel{=}{P} \;\; \uplus\;\; 1.2\stackrel{=}{N} \;\;\uplus,\;\; 1.2\stackrel{=}{O} \;\; \uplus \;\;1.2\stackrel{=}{U} \;\;\uplus\;\; 1.2\stackrel{=}{E} \;\; \uplus\;\; 1.2\stackrel{=}{D} \;\; \uplus\;\; 1.2\stackrel{=}{T} \;\; \uplus \;\;1.2\stackrel{=}{S} \;\; \uplus\;\; 1.2\stackrel{=}{F} )$

If we reverse the order of the equation, we have a new name and a different equation.

$2.1\stackrel{=}{=} \;\; \blacktriangleleft \;\; Ido(O,U,E,D,T,S,F,X) == Ori(O,U,E,D,T,S,F,X)$

This equation is just as valid as Equation $1.2\stackrel{=}{=}$ , As are all the subcases like Equation 2.1\stackrel{=}{M} , Equation 2.1\stackrel{=}{P} </m> , etc.

$1.2\stackrel{=}{=} \ldots 2.1\stackrel{=}{=} (1.2\stackrel{=}{=}) \ldots (2.1\stackrel{=}{=})$

These identity equations are usually commutative so it does not matter if the Ori, Ido or Cito part comes first. For example $Cito(\ldots ) =\!\!m\!\!= Ori(\ldots )$ will be the same as$Ori(\ldots ) =\!\!m\!\!= Cito(\ldots )$ . As a general rule we will start with the lowest numeral first but the reverse will be valid also.

The Principle of Name Interchangeability or PNI is that the name of a object or concept can be changed without changing the concept of object. If a term is being used as a name and not a functional operator and if renaming does not effect the concept or any aspect of the concept, renaming by another term is allowed. A term is used as a name and not a functional operator when the functional operator acts as a name and does not effect the terms it applies to in that situation.

## Simple Citoequations

\neq

$\neq$

$1.4\stackrel{=}{=} \; \; \blacktriangleleft \; Ori(O,U,E,D,T,S,F,X) == Cito(O,U,E,D,T,S,F,X)$ is an oriequation because it starts with a numeral 1 or "ori" in the elaboration.

The citoequation is actually written $4.1\stackrel{=}{=} \; \; \blacktriangleleft\;\;Cito(O,U,E,D,T,S,F,X) == Ori(O,U,E,D,T,S,F,X)$

$1.1\stackrel{=}{U} \; \;\blacktriangleleft \; \;Ori(O,U,E,D,T,S,F,C,X) =\!\!U\!\!= Ori(O,U,E,D,T,S,F,C,X)$

This both equation means the same thing just stated in reverse order. what the equation means that there is two different names for the same thing. This is because all of the factors in the parenthesis are the same. This follows from the equation 1.1=, and the principle of name replacement PNI.

$4.1\stackrel{=}{P} \; \; \blacktriangleleft \; \; Cito(O,U,E,D,T,S,F) =P= Ori(O,U,E,D,T,S,F)$

They are physically identical because they are the same thing. They just have a different name. They are the same because their indices are the same. This follows from $\; \; 1.1\stackrel{=}{=}$ and PNI. Or as a subcase of Equation $1.4\stackrel{=}{=}$ .

$1.1\stackrel{=}{U} \; \;\blacktriangleleft \; \; Ori(O,U,E,D,T,S,F,C,X) =\!\!U\!\!= Ori(O,U,E,D,T,S,F,C,X)$

$4.1\stackrel{=}{M} \; \;\blacktriangleleft \; \;Cito(O,U,E,D,T,S,F) =\!\!M\!\!= Ori(O,U,E,D,T,S,F)$

This means that the actual material bodies are mentally identical. Again this may be viewed as a tautology because they are the same thing but with a different name. They are of the same original, in the same universe, are made of the same matter, in the same place, and time with exactly the same structure and functioning. If we wish to be precise, the meanings the concept of a Cito(\ldots ) is different from the concept of a Ori(...). We have two different concepts applied to the same thing -- (O,U,E,D,T,S,C).

$4.1\stackrel{=}{F} \; \;\blacktriangleleft \; \; Cito(O,U,E,D,T,S,F) =\!\!F\!\!= Ori(O,U,E,D,T,S,F)$

This equation is true when and because$Ori(F) == Cito(F)$. This is correct because the functioning F is the same in both terms. In this case we have the same thing with a different name. It also follows from 1.1.F and principle of name interchangeability PNI. It also follows as a subcase of 1.4.=.

$4.1\stackrel{=}{B} \; \;\blacktriangleleft \; \; Cito(O,U,E,D,T,S,F) =\!\!B\!\!= Ori(O,U,E,D,T,S,F)$

It is hard to argue with the idea that a person will act exactly like itself when all factors are the same except for its name Cito instead of Ori. This follows from equation 1.1.B and the principle of name interchangeability (PNI).

So far these equations have been a little redundant because it defines exactly the same thing in each case. The only difference is in the names. These equations are valid and they begin to show the logic of this field of science and its mathematics.

In the following equations we will not always include the equations for the relationship between the original and idoriginal. They are essentially sub cases of the equations for the original and cidentireplica.

$4.1\stackrel{=}{=} \supseteq [4.1M \ \uplus 4.14 \ \uplus 4.1P \ \uplus 4.1/N \ \uplus 4.1O \ \uplus 4.1U \\uplus 4.1E \uplus 4.1D \ \uplus 4.1T \ \uplus 4.1S \ \uplus 4.1F]$

This is just a restatement of definition of 1.4= in mathematical terms.

$4.1\stackrel{=}{=} \ldots$states that if $1.4\stackrel{=}{=}$ is valid then the equations $4.1M,1.4 B, 4.1P, 1.4N, 4.1O, 4.1U, 4.1E, 4.1D, 4.1T, 4.1S, and 4.1F$ are valid as well. How is $4.1N$ valid when $Ori(O,U,E,D,T,S,F) =\!\!N\!\!= Cito(O,U,E,D,T,S,F)$ is not true? This is true because of the principle of name interchangeability.

$(1.4\stackrel{=}{=} \; \; \supseteq 4.1\stackrel{=}{=} ) \; \; \supseteq \; \;(1.4\stackrel{=}{=} ) \; \; \supseteq \; \;(4.1\stackrel{=}{=} )$

$1.4\stackrel{=}{=} 4.1\stackrel{=}{=} \; \;\supseteq \; \; (1.4\stackrel{=}{=} \; \;\supseteq) \supseteq \; \;(4.1\stackrel{=}{=} \; \; \supseteq)$

$1.4\stackrel{=}{=} \; \; \supseteq \; \; (1.4\stackrel{=}{M} \;\; \uplus\;\; 1.4\stackrel{=}{B} \;\; \uplus\;\; 1.4\stackrel{=}{P} \;\; \uplus\;\; 1.4\stackrel{=}{N} \;\;\uplus,\;\; 1.4\stackrel{=}{O} \;\; \uplus \;\;1.4\stackrel{=}{U} \;\;\uplus\;\; 1.4\stackrel{=}{E} \;\; \uplus\;\; 1.4\stackrel{=}{D} \;\; \uplus\;\; 1.4\stackrel{=}{T} \;\; \uplus \;\;14\stackrel{=}{S} \;\; \uplus\;\; 1.4\stackrel{=}{F} )\supseteq\; \;

4.1\stackrel{=}{=} \; \; \supseteq \; \; (4.1\stackrel{=}{M} \;\; \uplus\;\; 4.1\stackrel{=}{B} \;\; \uplus\;\; 4.1\stackrel{=}{P} \;\; \uplus\;\; 4.1\stackrel{=}{N} \;\;\uplus,\;\; 4.1\stackrel{=}{O} \;\; \uplus \;\;4.1\stackrel{=}{U} \;\;\uplus\;\; 4.1\stackrel{=}{E} \;\; \uplus\;\; 4.1\stackrel{=}{D} \;\; \uplus\;\; 4.1\stackrel{=}{T} \;\; \uplus \;\;4.1\stackrel{=}{S} \;\; \uplus\;\; 4.1\stackrel{=}{F} )$

This is an equation that shows the associative principle in the nature of these equations.

## Reduction of terms rules

$1.1\stackrel{=}{=}\stackrel{F}{(F)} \; \;\blacktriangleleft \;\; F(Ori(O,U,E,D,T,S,F)) == Ori(F)$

$1.4F \;\; \blacktriangleleft \;\; Ori(O,U,E,D,T,S,F) =\!\!F\!\!= Cito(O,U,E,D,T,S,F)$

This equation states that the functioning of the original with all its terms is identical to the originals functioning.

$1.1\stackrel{=}{F}(F) \; \;\blacktriangleleft \;\; Ori(O,U,E,D,T,S,F) =\!\!F\!\!= Ori(F)$

This equation says that the original with these terms (O,U,E,D,T,S,F) is functionally identical to the originals functioning.

Other Equations of Reduction

$1.1OO\supseteq, \; \; \blacktriangleleft \; O(Ori(O,U,E,D,T,S,C)) \supseteq Ori(O)$

$1.1\stackrel{=}{O}(O) \;\blacktriangleleft \; Ori(O,U,E,D,T,S,C) =\!\!O\!\!= Ori(O)$.

$1.1\stackrel{=}{U}(U)\; \;\supseteq\; \; 1.1U(U)$

$1.1\stackrel{=}{D}(D)\; \;\supseteq \; \;1.1D(D)$

$1.1\stackrel{=}{T}(T)\; \; \supseteq \; \;1.1(T)$

$1.1=S(S) \; \; \supseteq \; \;1.1S(S)$

Equations using functional operators

$1.1\stackrel{=}{M}\stackrel{\ 2}{M} \; \blacktriangleleft \; M{Ori(O,U,E,D,T,S,F)} =\!\!M\!\!= M{Ori(O,U,E,D,T,S,F)}$

Equation $1.1\stackrel{=}{M}\stackrel{\ 2}{M}$ , means that the original's mentality, is mentality equal to it own mentality when the two have the same identical original (O), Are in the same universe with the same universal physical laws (U),are made of the same matter/energy (E ), in the same dimensions/space (D), in the same time (T) , have identical structure (S),and functioning identically (C) . In this case the superscript “1” in “1.2” represents one relational operator and the “2” represents 2 functional operators.

$1.2\stackrel{=}{M}\stackrel{\ 2}{M} \; \blacktriangleleft \; M(Ori(O,U,E,D,T,S,F)) =\!\!M\!\!= M(Ido(O,U,E,D,T,S,F))$

### Reduction rules for Itoequations

#### Functional operator reduction

A specific example:

$F(F[Ori(O,U,E,D,T,S,C)]) \;\; \supseteq \;\;F[Ori(O,U,E,D,T,S,F)]$

A general example:

$F(F[Ito(\ldots)]) \;\; \supseteq \;\;F[Ito(\ldots)]$

Multiple simultaneous reduction of Functional operators:

$p(F)(F[Ito_i(O,U,E,D,T,S,C)]) \;\; \supseteq \;\;F[Ito_i(O,U,E,D,T,S,F)]$

$Ito_i$ represents any type of itobody such as an original, cidentireplica, or simidentireplica.

#### Relational operator reduction

A specific example:

$Ori_1(O,U,E,D,T,S,F) =\!\!F\!\!= =\!\!F\!\!= Ori(1)(O,U,E,D,T,S,F)\;\; \supseteq \;\;Ori(1)(O,U,E,D,T,S,F) =\!\!F\!\!= Ori(1)(O,U,E,D,T,S,F)$

A general example:

$Ito(\ldots) =F= =F= Ito(\ldots) \;\; \supseteq \;\; Ito(\ldots) =F= Ito(\ldots)$

Multiple simultaneous reduction of Relational operators:

$Ito_i(O,U,E,D,T,S,F)\!\! m(=\!\!F\!\!=) \!\! Ito_i(O,U,E,D,T,S,F)\;\; \supseteq \;\;Ito_i(O,U,E,D,T,S,F) =\!\!F\!\!= Ito_i(O,U,E,D,T,S,F)$

rules of parentheses reduction or addition

(=m=) \supseteq =m=</m>

$((O,U,E,D,T,S,F)) \supseteq (O,U,E,D,T,S,F)$

#### Term reduction

A specific example:

$Ori_1(O,U,E,D,T,S,F,F)\;\; \supseteq\;\; Ori(1)(O,U,E,D,T,S,F)$

A general example:

$Ito(F,F) \;\; \supseteq\;\; Ito(F)$

Multiple simultaneous reduction of terms

$Ito_i(n_2O,n_3U,n_4E,n_5D,n_6T,n_7S,n_8F)\;\; \supseteq\;\; Ito_i(O,U,E,D,T,S,F)$

#### Multiple simultaneous reduction of Terms, and Functional operators

A specific case:

$p(F)(F[Ito_i(n_2O,n_3U,n_4E,n_5D,n_6T,n_7S,n_8F)])\;\; \supseteq\;\; F[Ito_i(O,U,E,D,T,S,F)]$

A general case:

$p(\mathfrak{f})(\mathfrak{f}[Ito_i(n_2O,n_3U,n_4E,n_5D,n_6T,n_7S,n_8F)])\;\; \supseteq\;\; \mathfrak{f}[Ito_i(O,U,E,D,T,S,F)]$

#### Multiple simultaneous reduction of Terms, Relational operators, and Functional operators:

$p(\mathfrak{f})(\mathfrak{f}[Ito_i(n_2O,n_3U,n_4E,n_5D,n_6T,n_7S,n_8F)]) \;\; m(=\!\!\mathfrak{R}\!\!=) \;\; \mathfrak{f}[Ito_i(O,U,E,D,T,S,F)] \;\; \supseteq\;\;p(\mathfrak{f})(\mathfrak{f}[Ito_i(n_2O,n_3U,n_4E,n_5D,n_6T,n_7S,n_8F)]) \;\; (=\!\!\mathfrak{R}\!\!=) \;\; \mathfrak{f}[Ito_i(O,U,E,D,T,S,F)]$

$p(\mathfrak{f})(\mathfrak{f}[Ito_i(n_2O,n_3U,n_4E,n_5D,n_6T,n_7S,n_8F)]) \;\; m(=\!\!\mathfrak{R}\!\!=) \;\; \;\; \supseteq\;\; (=\!\!\mathfrak{R}\!\!=) \;\; \mathfrak{f}[Ito_i(O,U,E,D,T,S,F)]$

$\mathfrak{f}$ is a grouping symbol for all of the different kinds of functional non recursive operators

$\mathfrak{R}$ is a grouping symbol for all of the different kinds of relational non recursive operators.

#### Expansion rules

1. Functional operator reduction (non recursive)

F[Ori(O,U,E,D,T,S,C)] e F{ F[Ori(O,U,E,D,T,S,C)]}

K is a non recursive knowledge functional operator

K[Ori(O,U,E,D,T,S,C)] e K{ K[Ori(O,U,E,D,T,S,C)]}

Not a valid for certain functional operators like knowledge K.

K is a recursive knowledge functional operator

K[Ori(O,U,E,D,T,S,C)] /… K{ K[Ori(O,U,E,D,T,S,C)]}

When a functional operator can be applied to itself and” the resulting product is not itself it is a recursive functional operator. The symbol “/…” means does not imply

2. Relational operator reduction

Ori_1(O,U,E,D,T,S,C) =F= =F= Ori_1(O,U,E,D,T,S,C) e

Ori_1(O,U,E,D,T,S,C) =F= Ori_1(O,U,E,D,T,S,C)

Relational operator expansion

Ori_1(O,U,E,D,T,S,C) =F= Ori_1(O,U,E,D,T,S,C) e

Ori_1(O,U,E,D,T,S,C) =F= =F= Ori_1(O,U,E,D,T,S,C)

3. Term reduction

Ori_1(O,U,E,D,T,S,C,C) e Ori_1(O,U,E,D,T,S,C)

Term expansion

Ori_1(O,U,E,D,T,S,C) e Ori_1(O,U,E,D,T,S,C,C)

The concept of name reduction this is where when all other factors but the name is the same

Por_a(U,,S,C) == Por_b(U,,S,C)

Por represents a potential original or an original that does not exist in space or time but has the potential of existence.

Pci(a)(U,,S,C) ==Pci(b)(U,,S,C)

Pci represents a potential cidentireplica or a cidentireplica that does not exist in space or time but has the potential of existence.

Aor(a)(U,,S,C) == Aor(b)(U,,S,C)

Aor represents a actual original that exists somewhere in space or time.

Aci(a)(U,,S,C) == Aci(b)(U,,S,C)

Aci represents a actual cidentireplica that exists somewhere in space or time.

## Functional Equations

$1.2\stackrel{=}{M} {_M^{2()}} \; \blacktriangleleft \; M(Ori(O,U,E,D,T,S,F)) =\!\!M\!\!= M(Ido(O,U,E,D,T,S,F))$

Equation $1.2\stackrel{=}{M}{_M^{\ 2}}$ means that an original’s mentality is mentally equal to the mentality of its idoriginal when the two have the same identical original (O), Are in the same universe with the same physical laws (U), are made of the same matter (E), in the same dimension/space (D), in the same time (T) , have identical structure (S), and functioning identically (C) . This is the same equation as above with a name replacement Ori instead of Ido.

$1.4\stackrel{=}{M} {_M^{2()}} \; \;\blacktriangleleft \; \; M(Ori(O,U,E,D,T,S,F)) =\!\!M\!\!= M(Cito(O,U,E,D,T,S,F))$

Equation $1.4\stackrel{=}{M} {_M^{\ 2}}$ means that the mentality of the cidentireplica is mentally equal to the mentality of the original when the two have the same identical original (O), exists in the same universe with the same physical laws (U), are made of the same matter (E), in the same dimension/space (D), in the same time (T), have identical structure (S), and are functioning identically (C) .

$1.1\stackrel{=}{F} {_F^{2(}} \; \; \blacktriangleleft \; \; F(Ori(O,U,E,D,T,S,F)) =\!\!F\!\!= F(Ori(O,U,E,D,T,S,F))$

The functioning of the original is functionally equivalent to the functioning of the original. The F 1.2 in the equation designation means that there is one relational operator and two functional operators in the equation. If there are two relational operator in the equation then we represent it as F 2.. F 3. for three relational operators.

An example of more relational and functional operators in an equation is:

$1.1_{F}^{\stackrel{2}{=}} {_F^{3(}} \; \;\blacktriangleleft\; \; F(Ori(O,U,E,D,T,S,F)) =\!\!F\!\!= F(Ori(O,U,E,D,T,S,F)) =\!\!F\!\!= F(Ori(O,U,E,D,T,S,F)) =\!\!F\!\!= F(Ori(O,U,E,D,T,S,F))$

In this equation there are three relational operators and four functional operators

Equation 1.1\stackrel{=}{F}_F^2 \blacktriangleleft Ori(F) =\!\!F\!\!= Ori(F)

$1.1\stackrel{=}{F} {_F^{2(}} \; \; \blacktriangleleft \; \; F(Ori(O,U,E,D,T,S,F)) =\!\!F\!\!= F(Ori(O,U,E,D,T,S,F))$

$1.4F \; \; \blacktriangleleft \; \; Ori(O,U,E,D,T,S,F) =\!\!F\!\!= Cito(O,U,E,D,T,S,F)$

The originals functioning is functionally equivalent to the originals functioning.

This is correct because the functioning F is the same in both terms. The term (F)2 is used to represent the double use of C.

Equation $1.1_F^2_F^3 \; \; \blacktriangleleft \; \; Ori_(F) =\!\!F\!\!= Ori_F =\!\!F\!\!= Ori_F$

In this equation there are two relational operators .F2 And three references to functioning (F)3. Equation $1.1F_nF_m$ extends this relationship to any number n, where m will be n + 1.

Equation 1.4B_B^1 B(Ori(O,U,E,D,T,S,F)] =\!\!B\!\!= Cito(O,U,E,D,T,S,F)

$1.4\sideset{_{1.1}^B}{}{B} \; \; \blacktriangleleft \; \; B(Ori(O,U,E,D,T,S,F)) =\!\!B\!\!= Cito(O,U,E,D,T,S,F)$

This equation states that the behavior of the original is behaviorally identical to the cidentireplica. We use the superscript B1.1. To define that there is one relational operator =B= and one functional operator B[ ...] in the equation.

Equation 1.4.B1.2. ﬁ B(Ori(O,U,E,D,T,S,F)] =\!\!B\!\!= B(Cito(O,U,E,D,T,S,F))

$1.4\sideset{_2^B}{}{B} \; \; \blacktriangleleft \; \; B(Ori(O,U,E,D,T,S,F)] =\!\!B\!\!= B(Cito(O,U,E,D,T,S,F))$

B1.2 means that there are one relational operator and two functional operators in this equation.

$1.4B{_F^{2(}}$ \stackrel{=}{B} {_F^{2(}}

$1.4\sideset{_{2F}^B}{}{B} \; \; \blacktriangleleft \; \; F(Ori(O,U,E,D,T,S,F)) =\!\!B\!\!= F(Cito(O,U,E,D,T,S,F))$

The functioning of the original is not the original or the behavior of the original.

This equation states that the functioning of the original is behaviorally equal to functioning of the cidentireplica. Technically since there is no behavior i.e., “the null behavior” and both sides have this null behavior so they are by default behaviorally equal.

Equation $1.4=_F^2 \; \; \blacktriangleleft \; \; F(Ori(O,U,E,D,T,S,F)) == F(Cito(O,U,E,D,T,S,F))$

$1.4F \; \; \blacktriangleleft \; \; Ori(O,U,E,D,T,S,F) =\!\!F\!\!= Cito(O,U,E,D,T,S,F)$

This states that the functioning of the original is identical to the functioning of the cidentireplica. But the double equal sign also means identical in all other ways defined. Such as =M=, =E= , etc. This is true because these other relational operators will be relating null or empty theocepts and thus will be identical.

$1.4F…B$

$1.4F \ldots 1.4B \;\; \blacktriangleleft \;\; Ori(O,U,E,D,T,S,F) =\!\!F\!\!= Cito(O,U,E,D,T,S,F) \;\ldots\; Ori(O,U,E,D,T,S,F)) =\!\!B\!\!= Cito(O,U,E,D,T,S,F)

This equation states that if the [[original]] and the [[cidentireplica]] are functionally identical then they will be behaviorally equal. In the case where there is not behavior produced it is true by default.

==Mixed Equations ==

\equiv \not\equiv \ne \mbox{or} \neq \propto

\equiv \not\equiv \ne \mbox{or} \neq \propto">

$1.4\sideset{_{2F}^B}{}{B} \blacktriangleleft F[Ori(O,U,E,D,T,S,C)] =\!\!B\!\!= F[Cito(O,U,E,D,T,S,C)]$

New relational operators for equations

R14: $\not\equiv$ means not equal or identical, in one or more ways but not in all ways. We can put the slash sign in front of any relational operator for example;$=\!\!/B\!\!=,=\!\!/P\!\!=,=\!\!/M\!\!=, =\!\!/S\!\!=$ etc.

R15: $\ne\ne$. This relational operator means the terms are not equal in all ways.

For terms to be different there has to be a difference in the terms. To represent this difference we have added subscripts and superscripts to the terms, and to the terms in the names of the equations.

Equation 1.1\stackrel{E}{\not\equiv}1^m) Ori(O,U,E1,D,T,S,C) =\!\!/\!\!=Ori(O,U,Em,D,T,S,C)

$1.1\sideset{}{_{1}^E}{=/=} \blacktriangleleft Ori(O,U,E_1,D,T,S,C) =\!\!/\!\!= Ori(O,U,E_m,D,T,S,C)$

These are not identical because they are made of different matter. E1 does not equal Em . Of course they are still in the same space and time, which can cause some problems, if this situation could actually occur. In this case we have the same name for two things that are different in one way. In reality the original could have been made of different matter.

Equation 1.1/P(E1m) Ori(O,U,E_1,D,T,S,C) =\!\!/P\!\!= Ori(O,U,E_m,D,T,S,C)

$1.1\sideset{_{2F}^B}{}{B} \xleftarrow{Compaction} \dot= \xrightarrow{Elaboration} Ori(O,U,E_1,D,T,S,F) =\!\!/P\!\!= Ori(O,U,E_m,D,T,S,F)$

The original is not physically equal to itself when it is made of different matter. The term E1m defines the change in matter from E1 to Em. E1 is fixed to 1 a specific grouping and arrangement of matter, but Em is a variable because m represents a variable. So Em represents any grouping or arrangement of matter that satisfies the other conditions (O,U,D,T,S,C).

Equation 1.4./(E_1m) ﬁ Ori(O,U,E_1,D,T,S,C) =/= Cito(O,U,E_m,D,T,S,C)

$1.4\sideset{}{_{1}^E}{=/=} \blacktriangleleft Ori(O,U,E_1,D,T,S,F) =/= Cito(O,U,E_m,D,T,S,F)$

Like equation 1.1./(E1m) the cidentireplica and the original are not identical because they are not made of the same matter but they have the peculiar situation of being in the same place and at the same time. Whether this is physically possible is another question.

There are a number of equations that are not equivalent. There are how many equations using the relational operator =//=? In equation 1.4.// we do not need parenthesis because it includes all terms. But this equation is false because every term on both side of the equation is identical so they are equal.

1.4.//(1m) ﬁ Ori(O1,U1,E1,D1,T1,S1,C1) =//= Cito(Om,Um,Em,Dm,Tm,Sm,Cm)

$1.4\sideset{_{2F}^B}{}{B} \blacktriangleleft Ori(O_1,U_1,E_1,D_1,T_1,S_1,F_1) =//= Cito(O_m,U_m,E_m,D_m,T_m,S_m,F_m)$

This equation will be valid because each term on different sides of the equation are different. (1m) represents all terms transposing from 1 in the original to m in the cidentireplica. However in this equation we cannot, by definition, call the cidentireplica a cidentireplica of this original because it is not identically functioning.

1.4.//(n m) Ori(O_n,U_n,E_n,D_n,T_n,S_n,C_n) =//= Cito(O_m,U_m,E_m,D_m,T_m,S_m,C_m)

What this equation states is that the original and the cidentireplica are not identical in any way defined by these terms.

1.3.//(1m)… ﬁ 1.3.//(1m)…1.3./N(1m) ﬁ

1.3.//(1m) … 1.3./B(1m) » 1.3./M(1m) » 1.3./P(1m) » 1.3./N(1m) 1.3./S(1m) » 1.3./F(1m) » 1.3./O(1m) » 1.3./U(1m) » 1.3./E(1m) » 1.3./D(1m) » 1.3./T(1m) This equation can be elaborated again.

## Symbols for the construction of equation

\sideset{_1^2}{_3^4}\prod_a^b

$\sideset{_1^2}{_3^4}\prod_a^b$

{}_1^2\!\Omega_3^4

${}_1^2\!\Omega_3^4$

\overset{\alpha}{\omega}

$\overset{\alpha}{\omega}$

\underset{\alpha}{\omega}

$\underset{\alpha}{\omega}$

\overset{\alpha}{\underset{\gamma}{\omega}}

$\overset{\alpha}{\underset{\gamma}{\omega}}$

\stackrel{\alpha}{\omega}

$\stackrel{\alpha}{\omega}$

\forall \exists \empty \emptyset \varnothing

$\forall \exists \empty \emptyset \varnothing$

\in \ni \not \in \notin \subset \subseteq \supset \supseteq

$\in \ni \not \in \notin \subset \subseteq \supset \supseteq$

\cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus

$\cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus$

\sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup

$\sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup$

$\blacktriangleleft$

$\blacktriangleright$

\varpropto \blacktriangleleft \therefore \backepsilon \blacktriangleright \because \nleqslant \nleqq \lneq \lneqq

$\varpropto \blacktriangleleft \therefore \backepsilon \blacktriangleright \because \nleqslant \nleqq \lneq \lneqq$

\varpropto \blacktriangleleft \therefore \backepsilon \blacktriangleright \because \nleqslant \nleqq \lneq \lneqq

$\varpropto \blacktriangleleft \therefore \backepsilon \blacktriangleright \because \nleqslant \nleqq \lneq \lneqq$

\sideset{_a^b}{_c^d}M

\sideset{_a^b}{_c^d}a

∈ ∉ ∩ ∪ ⊂ ⊃ ⊆ ⊇ _{\models}^=_F^{\ 2}

$_{\models}^=_F^{\ 2}$

$_F^{\ 2}$

$1.4 {\overset{M}{M}M} \blacktriangleleft M(Ori(O,U,E,D,T,S,F)) =M= M(Cito(O,U,E,D,T,S,F))$

$1.4 \sideset{_a^b}{_c^d}{M} \blacktriangleleft M(Ori(O,U,E,D,T,S,F)) =M= M(Cito(O,U,E,D,T,S,F))$

$1.4\sideset{_a^b}{_c^d}M \blacktriangleleft M(Ori(O,U,E,D,T,S,F)) =M= M(Cito(O,U,E,D,T,S,F))$

$1.4 \sideset{_a^b}{}{M} \blacktriangleleft M(Ori(O,U,E,D,T,S,F)) =M= M(Cito(O,U,E,D,T,S,F))$

$1.4\sideset{_a^b}{}M \blacktriangleleft M(Ori(O,U,E,D,T,S,F)) =M= M(Cito(O,U,E,D,T,S,F))$

$1.4 \sideset{_M^M}{}{M} \blacktriangleleft M(Ori(O,U,E,D,T,S,F)) =M= M(Cito(O,U,E,D,T,S,F))$

$1.4\sideset{_M^M}{_c^d}M \blacktriangleleft M(Ori(O,U,E,D,T,S,F)) =M= M(Cito(O,U,E,D,T,S,F))$

$1.4M\_1^2\!\Omega_3^4 \blacktriangleleft M(Ori(O,U,E,D,T,S,F)) =M= M(Cito(O,U,E,D,T,S,F))$