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Abstract

Cognitive processes of human brain are strongly tied to such a well-known part of the brain as cortex. All psychological, logical or illogical solutions made by a human being are the result of the cortex. Thus, the maximum approximation of mathematical theory to the processes of the cortex can become a good trampoline to the creation of a self-learning intellectual system, a Real Artificial Intelligence.We propose a new concept of mathematical theory which gives a possibility to form, find and separate senses of two or more objects of different nature. The theory encompasses the knowledge of cybernetics, linguistics, neurobiology, and classical mathematics.The Sense Theory is not a part of traditional mathematics as we know it now, it is a new paradigm of how we can formalize complex cognitive processes of the human brain.
Sense Theory
(part 1)
[P-S Standard]
Egger Mielberg
egger.mielberg@gmail.com
22.04.2019
Abstract.
Cognitive processes of human brain are strongly tied to such a well-
known part of the brain as cortex. All psychological, logical or illogical
solutions made by a human being are the result of the cortex. Thus,
the maximum approximation of mathematical theory to the processes
of the cortex can become a good trampoline to the creation of a self-
learning intellectual system, a Real Artificial Intelligence.
We propose a new concept of mathematical theory which gives a
possibility to form, find and separate senses of two or more objects of
different nature. The theory encompasses the knowledge of
cybernetics, linguistics, neurobiology, and classical mathematics.
The Sense Theory is not a part of traditional mathematics as we
know it now, it is a new paradigm of how we can formalize complex
cognitive processes of the human brain.
1. Introduction
While the definition of artificial intelligence is unclear so far, we believe that
cognitive characteristic is the main and first step anyone who creates AI
should start from. This choice has one strong reason. Humans have five
traditionally recognized senses, sight (vision), hearing (audition), taste
(gustation), smell (olfaction), and touch (somatosensation). All the senses
generate data that the brain needs to perceive and comprehend. Thus,
mechanisms of data processing are crucial for such an important human
act as decision making. That is why we consider the fundamentally different
test in comparison with the Turing test.
Test of Three Persons.
The test consists of the following steps:
1. Three persons, Rick, Nick and Dick are taken. By arbitrary choice,
one of the persons is substituted by a computer program (digital
machine).
2. Three persons exchange their dates of birth and start joint
conversation during the next 30 minutes.
3. A text with three arbitrary dates and numbers is exposed to the
persons for reading.
4. Three persons start a conversation about the text during the next 30
minutes.
5. Each person is asked, “Who is the machine?” with one required
sentence of answer explanation.
6. After the exposition of all answers for the persons, they are secondly
asked, “Who wants to change the answer?”.
7. The answers of the persons are fixed and calculated. If the machine
(Rick, Nick or Dick) was not chosen by other two persons
simultaneously, the test is passed.
Remark: During the test, each person (man) can make notices.
Graphically it can be shown as follows:
The passed test means only that the machine is capable of thinking
intellectually. For the level of human thinking it needs to pass the test of N
persons. We called the test of three persons as a test of first order. And the
test of N persons as a test of second order. Importance of intellectual level
for the machine can be compared with a process of approximation. In our
context, the figure inside the machine circle has as many sides as the
number of persons.
So, an intellectual excellence of the machine can be reached as soon as
the internal machine’s figure becomes the circle with minimum error.
2. Problem
Classical mathematics, namely its basis, classical mathematical logic is not
capable of operating with such an object as a sense. In other words, with
the qualitative properties of the object. However, it is crucial for building a
self-learning system of any kind.
In mathematics, as we know it now, there are a number of direct proofs of
the foregoing statement. We will briefly describe here only two of them
which are the basic ones according to the author of this article.
The law of the excluded middle (third).
By simple words, according to this law, if statement A is true then
statement which is opposite to the statement A is always false.
For example:
Statement A - "Bob is a stupid man".
Statement B - "Bob is a smart man".
Statement C - "Bob is a good man".
Statement D - “Bob is a bad man”.
So, if someone states that Bob is a stupid man, according to the law of the
excluded middle Bob cannot be simultaneously a smart man. At first sight,
it seems logical but as soon as we list the characteristics (properties) of a
stupid man as well as a smart one we will necessarily bump into a
contradiction. As a matter of fact, part of the properties of the smart (stupid)
man can be the same one of any other (not stupid nor smart) man. In terms
of mathematical logic, we have:
  
or
  
where    without fail.
Further, if someone states state that Bob is a stupid but good man, we
have:
  
and
     
or
     
But in the practical realization of any intellectual system, we frequently
meet the situation when a man has several properties simultaneously or
property that does not have direct opposite value. For example, the
following expression cannot be firmly established or refuted:
       
or
               
where S statement “Bob is a shapely man”, G – statement “Bob is an
elegant man”. Thus, the classical mathematical logic is good only for
homogeneous objects that do not have qualitative properties.
Gödel’s incompleteness theorems.
Gödel’s theorems are a good example of the absence of a clear and single
definition of what "negation operation" is all about. In classical logic, it is
primarily used in the context of two possible values, "true" or "false". In this
way, only propositions that can be evaluated by two states are possible for
operation and analysis. Therefore, the negation operation is good if and
only if the outcome of any proposition can take two opposite forms.
One of the Gödel’s theorem says that we cannot derive two formulas
 simultaneously, where   .
But what exactly does  mean? Suppose we have the following
series of values:
    
Thus, all the above values are true. Then, ” should mean situation
when the values are false. In other words,  is undefined for   . For
example, if we take the following simple formula  ,   ,
then    . In case of “” means opposite value,
 . Thus, we have two formulas and two sets. It clearly shows that
Gödel’s theorems as well as classical logic (its operators) are primarily
focused on Boolean domain. In other words, it works only when a bijective
function is defined.
In the context of the Sense Theory as well as any practical realization of a
semantical live algorithm, there are more than two states for an object. For
example, the object “device” can have more than one qualitative properties
such as "plastic", "thin", etc. But in the context of sense, it is undefined if it
does not have a single property. In practice, the Sense Theory operates
multivalued functions.
Resuming above-said and what can be derived from it, all logical operators
of the classical logic are primarily suiting to bijective sets. But it is
absolutely not suited to the nature of cognitive processes as well as the
Sense Theory.
3. Solution
At the core of the theory lies an object which has a qualitative property (‘s).
The object can be the nature of any kind. For example, a word “device”
presents a template of some element with no relationship to any categorical
context. As soon as we prefix the word “medicine” to the word “device”, the
corresponding context becomes evident. In this case, the word “medicine”
is the qualitative property of the word “device”.
 ,
where object,
index quantity of object properties.
In case of prefixing more different-in-sense-words, we get the following
notation:
    
 
or
,
where total quantity of object properties.
In terms of linguistics, the property of any object can be any part of speech.
The object that does not have any properties is called zero object:
 
or
device
A Sense Set (SS) of the objects is a plurality thought of as a sense unit.
Let us consider the following several objects:
     ,
    ,
    ,
    ,
  
In the next step, we will consider the all above-mentioned objects as
properties:
     
This set of properties forms a No-Sense Set (NS):
As it was said earlier, an object has a qualitative property (‘s). But some of
them can be a zero ones with no properties at all.
Now, if we start to select a zero object iteratively, with high probability we
will end up with the object “aircraft” or “airplane”.
 ,
and
or
where S Sense Set.
Unlike zero object, Sense Set cannot be empty as it is a result of “inclusion”
of two elements, zero object and No-Sense Set.
Definition 1: S is a Sense Set if and only if the following expression is true:
where N,K = {1,2,3,…n}, K N, K,N finite numbers.
Definition 2:
where N finite number.
Definition 3: is a zero or empty object if and only if the following
expression is true:
Definition 4:
where N finite number of objects.
Definition 5:
Definition 6:
where
Definition 7:
Definition 8:
Definition 9: Object A semantically connects to Object B if the following
expression is true:
Semantic connection” (SC) – is measured by percent. The following
formula is used:
  
where number of similar properties of both objects,
number of properties of largest object.
In order to formulate the first Axiom of the Sense Theory, we need to enter
such definitions as “sense sequence” and “sense limit”.
Definition 10: The set A of      elements is a sense sequence if
and only if there is at least a single zero object that satisfies the
following expression:
Definition 11: The sense limit of the set A is the zero object of a Sense Set.
In other words, an object the properties of which are the elements of the set
A is a sense limit of that set.
It has the following notation:
where      .
For the subset of the set A, we can have two outcomes:
or
The Axiom of Object Limit:
Every object of Sense Set consists of two parts, zero object and sense
sequence, where the first one is a sense limit of the second one.
The following two expressions are equivalent:
The Axiom of Object Equality:
Object is equal to Object if the following expression is true:
or
The Axiom of Set Equality:
The Sense Set is equal to the Sense Set if the following expression is
true:
or
The Axiom of Semantic Union (left-to-right):
1. For any two No-Sense Sets,
and there is such the properties of which are
both, the properties of and :
where K = M + L.
2. For any two Objects,  and , there is such  that the
following two expressions are true:
3. For any two Sense Sets,  and  there is such  that the
following two expressions are true:
The Axiom of Semantic Subset:
Any Object can be only one of two types of subsets for any Sense Set
:
Subset of first order:
where K = N.
Subset of second order:
where K = N or K N.
The Axiom of Set of Subsets:
“There are at least N+1 subsets for any .”
Theorem (Existence of Set).
“The Sense Set () is defined if and only if there is a sense limit of
.
Proof.
For any given there are two elements, and by definition. Now,
presume that there is no sense limit of . In symbolic notation, it
presents the following:
and
The latter expression contradicts the definition of Sense Set. The theorem
is proven.
Theorem (Existence of Subsets).
“There is at least one subset for and N+1 subsets for .
Proof.
Further,  
 
 
  
  
The theorem is proven.
4. Conclusion
In this article, we presented the new mathematical theory with own
signature. Unlike classical mathematical or intuitionistic logic, the Sense
Logic which is the basis for the Sense Theory can drastically improve
understanding methods and possible algorithms in the creation of human-
like AI.
We hope that our decent work will help other AI researchers in their life
endeavors.
To be continued.
Appendix
“Semantic Intersection” is commutative for all operands.
Definition 9.
Associativity (“inclusion”):
Associativity (“semantic union”):
Associativity (“semantic disunion”):
References
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