An axiom system for dialectical logic K-model which is based on energy-method is established by author in the paper. Dialectical logic K-model supplies a computation-idea for machine, so that it can be applied in many computations for artificial intelligence. This paper described that subjective-laws is the mirror imagine reflected from objective-laws, and defined the three-step which is named by sensation, abstraction and t hinking separately in artificial intelligence. At the same time, the author described axiom system for dialectical logic K-model which contains logic-variable energy conservation law, Mozi-principle (minimax principle) and forbidden law, etc. In the axiom system, it also contains a continuous-true-value function valued on interval [-1, 1], and the K-graph for logic-variable. Dialectical logic K-model would supply a computation-idea for machine so that the machine is able to think by dialectical logic method, thus an important information-treated method may be the dialectical logic.
Objective world consists of five factors which are MATTER, ENERGY, INFORMATION, SPACE and TIME. Among these factors, there are many inner or outer laws of causation called objective laws, which can also be called as objective logic. Corresponding to objective laws, there is a mirror image in machine of artificial intelligence, which can be called subjective laws, and also can be called as subjective logic. The machine of artificial intelligence only holds objective laws through subjective laws, or with another word, only holds objective logic through subjective logic. Subjective logic is built on infinite times circulation by the three-step named Sensation, Abstraction and Thinking. Sensation is input-ability to machine from objective information with sensor. Abstraction is conception-classification ability to sensed information and to name it. Thinking is the ability of machine researching from conception-classification information to get the subjective logic.
Dialectical logic K-model would supply a computation-idea for machine so that the machine is able to think by dialectical logic method [
The conception is the name what be named the object-thing’s main property and neglected its secondary property. Conception is the basis of artificial intelligence.
The conception-dimension is the dimension-number of conception. For example, HORSE is an one-dimension conception; WHITE HORSE is a two-dimension conception; RUNNING WHITE HORSE is a three-dimension conception; ONE RUNNING WHITE HORSE is a four-dimension conception; etc.
The three main identification technologies will setup three data-base, although there are other data-base, for example physical data-base and chemical data- base, etc. These three data-bases are associated by the conception-dimension.
Denote objective-demain by B, and subjective-demain by A; define research- arithmetic ∇ (research-arithmetic is an arithmetic to get subjective-logic from objective-logic) and inverse research-arithmetic ∇ − 1 (inverse research-arithmetic is an arithmetic to make subjective-logic to objective-logic to check its true or false) and thus
∇ ( B i ∈ B ) ⇔ ∇ ( A i ∈ A ) ∇ − 1 ∇ ( B i ∈ B ) ⇔ ∇ − 1 ∇ ( A i ∈ A ) ⋯ ∇ − 1 ∇ ⋯ ∇ − 1 ∇ ( B i ∈ B ) ⇔ ∇ − 1 ∇ ⋯ ∇ − 1 ∇ ( A i ∈ A )
∇ − 1 ∇ = F 1 ⋯ ∇ − 1 ∇ ⋯ ∇ − 1 ∇ ∇ − 1 ∇ ︸ j couplesof ∇ − 1 ∇ = F j
Then F j ( B i ∈ B ) ⇔ F j ( A i ∈ A )
As above, F j is the j-th cycle research-arithmetic, obviously F j is a logic-transformation between demain A and demain B. Sometimes can denote F j by F. Symbol ⇔ is denote the object logic and the subject logic are mirror imagine each other.
Denote subjective-logic-variable by
A i ± n ( t ) = { a 1 ± ( t ) , a 2 ± ( t ) , ⋯ , a n ± ( t ) }
as above, t is time-variable, right superscript ± n denote the logic-variable contains n couples of contradicting-subvariable a i ± ( t ) , n is called the rank-number of A i ± n ( t ) , right subscript i denote the i-th of causation law.
T ( A i ± n ( t ) ) is denoted the true-valued-function of logic-variable A i ± n ( t ) , and T ( A i ± n ( t ) ) ∈ [ − 1 , 1 ] .
Denote the mirror-image in objective-demain of A i ± n ( t ) ∈ A by B i ( t ) ∈ B .
To every object logic-variable B i ( t ) ∈ B , there must always exist logic-transformation F, make
A i ± n ( t ) = F ( B i ( t ) )
To every correct logic-transformation F, always make
lim j → ∞ C | A i ± n ( t ) − B i ( t ) | = lim j → ∞ C | F j ( B i ( t ) ) − B i ( t ) | = 0
i.e. to every finite j < ∞ , error-function
E j ( t ) = | F j ( B i ( t ) ) − B i ( t ) | > 0
In the formulation as above C = + 1 , if j = 2 p + 1 ; or C = − 1 , if j = 2 p , p = 0 , 1 , 2 , ⋯
In fact, incorrect logic-transformation F * can bring the researching divergence. The faster is the convergence of F j , the brighter is the F j .
For logic-variable A i ± n ( t ) , its inner producing-energy E i is equal to the consumed work J i by its contradiction-function (see the 10.4 contradiction-function as below), i.e.
E i = J i
Logic-variable A i ± n ( t ) in changing, must be satisfied or must be satisfied asymptotically by that cost-function C ( A i ± n ( t ) ) is minimum and gain-function G ( A i ± n ( t ) ) is maximum. Denote the pure gain-function by L ( A i ± n ( t ) ) , i.e.
L ( A i ± n ( t ) ) = max ( G ( A i ± n ( t ) ) − min C ( A i ± n ( t ) ) )
or
lim t → ∞ L ( A i ± n ( t ) ) = max t → ∞ ( G ( A i ± n ( t ) ) − min t → ∞ C ( A i ± n ( t ) ) )
1) Last-time memory law
For time-sequence t 1 < t 2 < ⋯ < t k < ⋯ , denote the memory effect-weight-function of contradiction-subvariable a i ± ( t p ) from time t by R ( a i ± ( t p ) ) ,
R ( a i ± ( t p ) ) ∈ [ 0 , 1 ]
and R ( a i ± ( t k ) ) > R ( a i ± ( t j ) )
If and only if t k > t j . Memory prefer the a k ± ( t ) of the last time.
2) Importance memory law
For importance-sequence M ( a p ± ( t ) ) < M ( a q ± ( t ) ) < ⋯ < M ( a r ± ( t ) ) , and M ( a i ± ( t ) ) ∈ [ 0 , 1 ]
So the memory prefer the a r ± ( t ) , if and only if its M ( a r ± ( t ) ) is bigger.
3) Bigger probability memory law
For property event α 1 , α 2 , ⋯ , α k , their corresponding probability are
P ( α 1 ) , P ( α 2 ) , ⋯ , P ( α k ) ,
and
P ( α P ) > P ( α q )
so the memory prefer α p .
In the researching process the machine must have ability to self-programmable and self-correctable without the operations by human beings.
There are two kinds of logic: one is dialectical-logic corresponding to intelligence quotient and another is imagine-logic corresponding to emotional quotient.
The imagine-logic would be forbidden into machine, because the imagine-logic will make machine to emotional quotient so that the autonomous-mind will belong to the machine. The artificial intelligence with the autonomous-mind will not like to be “a tool” for human beings, so “a new creation” will be created, of cause this is harmful to human beings.
A i ± n ( t ) = lim n → ∞ { a 1 ± ( t ) , a 2 ± ( t ) , ⋯ , a n ± ( t ) }
For cycle research-arithmetic F j , make
A i ± n ( t ) = F j ( B i ( t ) )
1) (inductive) if as above is true to finite F k , k = 1 , 2 , ⋯ , j
2) (hypothesis) suppose as above is true to F k + 1 ,
3) (deductive) so that as above is true to all j < ∞ .
Proof: through finite to prove infinite, there always exists error-function based on the 2.3. research error alternating-convergence law, if the 6.2. (hypothesis) is false, then producing the false to the 6.3. (deductive). Thus the truth to every step of F j would be carefully checked.
Proof is over.
For A i ± r ( t ) , A i ± s ( t ) , A i ± k ( t ) , denote their true-valued-function by corresponding to T ( A i ± r ( t ) ) , T ( A i ± s ( t ) ) , T ( A i ± k ( t ) ) ,
Definition T ( A i ± s ( t ) ) ∪ T ( A i ± r ( t ) ) = max { T ( A i ± s ( t ) ) ; T ( A i ± r ( t ) ) } logic ∪ is satisfied by as below
Commutative law T ( A i ± s ( t ) ) ∪ T ( A i ± r ( t ) ) = T ( A i ± r ( t ) ) ∪ T ( A i ± s ( t ) ) associative law
( T ( A i ± s ( t ) ) ∪ T ( A i ± r ( t ) ) ) ∪ T ( A i ± k ( t ) ) = T ( A i ± s ( t ) ) ∪ ( T ( A i ± r ( t ) ) ∪ T ( A i ± k ( t ) ) )
Definition T ( A i ± s ( t ) ) ∩ T ( A i ± r ( t ) ) = min { T ( A i ± s ( t ) ) ; T ( A i ± r ( t ) ) } logic ∩ is satisfied by as below commutative law
T ( A i ± s ( t ) ) ∩ T ( A i ± r ( t ) ) = T ( A i ± r ( t ) ) ∩ T ( A i ± s ( t ) ) associative law
( T ( A i ± s ( t ) ) ∩ T ( A i ± r ( t ) ) ) ∩ T ( A i ± k ( t ) ) = T ( A i ± s ( t ) ) ∩ ( T ( A i ± r ( t ) ) ∩ T ( A i ± k ( t ) ) )
Logic hybrid arithmetic is satisfied as below
Distributive law
T ( A i ± s ( t ) ) ∪ ( T ( A i ± r ( t ) ) ∩ T ( A i ± k ( t ) ) ) = ( T ( A i ± s ( t ) ) ∪ T ( A i ± r ( t ) ) ) ∩ ( T ( A i ± s ( t ) ) ∪ T ( A i ± k ( t ) ) )
N [ T ( A i ± n ( t ) ) ] = ± 1 − T ( A i ± n ( t ) ) , if T ( A i ± n ( t ) ) > 0 is positive in front of the 1; if T ( A i ± n ( t ) ) < 0 is negative in front of the 1.
Denote N [ N [ ⋯ N [ A i ± n ( t ) ] ] ] ︸ p = N p [ A i ± n ( t ) ]
Arithmetic N is satisfied by as below idempotent law N p [ A i ± n ( t ) ] = A i ± n ( t ) , if p = 2 k ; N p [ A i ± n ( t ) ] = N [ A i ± n ( t ) ] , if p = 2 k + 1 ; k = 0 , 1 , 2 , ⋯
N [ T ( A i ± s ( t ) ∩ T ( A i ± r ( t ) ) ) ] = N [ T ( A i ± s ( t ) ) ] ∪ N [ T ( A i ± r ( t ) ) ] (1)
N [ T ( A i ± s ( t ) ) ∪ T ( A i ± r ( t ) ) ] = N [ T ( A i ± s ( t ) ) ] ∩ N [ T ( A i ± r ( t ) ) ] (2)
Proof:
N [ T ( A i ± s ( t ) ) ∩ T ( A i ± r ( t ) ) ] = ± 1 − ( T ( A i ± s ( t ) ) ∩ T ( A i ± r ( t ) ) ) = ± 1 − min { T ( A i ± s ( t ) ) ; T ( A i ± r ( t ) ) } = max { ± 1 − T ( A i ± s ( t ) ) ; ± 1 − T ( A i ± r ( t ) ) } = N [ T ( A i ± s ( t ) ) ] ∪ [ T ( A i ± r ( t ) ) ]
Formulation (1) proof is over.
N [ T ( A i ± s ( t ) ) ∪ T ( A i ± r ( t ) ) ] = ± 1 − ( T ( A i ± s ( t ) ) ∪ T ( A i ± r ( t ) ) ) = ± 1 − max { T ( A i ± s ( t ) ) ; T ( A i ± r ( t ) ) } = min { ± 1 − T ( A i ± s ( t ) ) ; ± 1 − T ( A i ± r ( t ) ) } = min { N [ T [ A i ± s ( t ) ] ] ; N [ T [ A i ± r ( t ) ] ] } = N [ T ( A i ± s ( t ) ) ] ∩ N [ T ( A i ± r ( t ) ) ]
Formulation (2) proof is over.
Remark 1: As above when time t is degenerated into a constant and destroy the contradictions in the formulations, then the formulations will be degenerated into the mathematical model of formal logic, i.e. Boolean algebra.
For logic arithmetic of true-valued-function T ( A i ± s ( t ) ) , T ( A i ± r ( t ) ) , as below there is a composition theorem:
If equation T ( A i ± s ( t ) ) − T ( A i ± r ( t ) ) = 0 exist root-set T = { t 1 , t 2 , ⋯ , t m } ∈ [ t 0 , t ] , and t 1 < t 2 < ⋯ < t m so that
max ( T ( A i ± s ( t ) ) ; T ( A i ± r ( t ) ) ) = max t ∈ [ t 0 , t 1 ] ( T ( A i ± s ( t ) ; T ( A i ± r ( t ) ) ) ) + max t ∈ [ t 1 , t 2 ] ( T ( A i ± s ( t ) ) ; T ( A i ± r ( t ) ) ) + max t ∈ [ t 2 , t 3 ] ( T ( A i ± s ( t ) ) ; T ( A i ± r ( t ) ) ) + ⋯ + max t ∈ [ t m , t ] ( T ( A i ± s ( t ) ) ; T ( A i ± r ( t ) ) )
as same as above, also
min ( T ( A i ± s ( t ) ) ; T ( A i ± r ( t ) ) ) = min t ∈ [ t 0 , t 1 ] ( T ( A i ± s ( t ) ) ; T ( A i ± r ( t ) ) ) + min t ∈ [ t 1 , t 2 ] ( T ( A i ± s ( t ) ) ; T ( A i ± r ( t ) ) ) + min t ∈ [ t 2 , t 3 ] ( T ( A i ± s ( t ) ) ; T ( A i ± r ( t ) ) ) + ⋯ + min t ∈ [ t m , t ] ( T ( A i ± s ( t ) ) ; T ( A i ± r ( t ) ) )
If equation T ( A i ± s ( t ) ) − T ( A i ± r ( t ) ) = 0 in which the possible augment root and complex root will be removed out.
If T ( A i ± s ( t ) ) − T ( A i ± r ( t ) ) ≠ 0 , so the root-set T = ∅ that max ( T ( A i ± s ( t ) ) ; T ( A i ± r ( t ) ) ) or min ( T ( A i ± s ( t ) ) ; T ( A i ± r ( t ) ) ) in which select a maximum or minimum of pure function.
Proof: obviously based on definition of logic arithmetic.
Proof is over.
Defines a connecting directed simple graph G K with 2 ( n + 1 ) nodes and E edges G K = { V 2 ( n + 1 ) ; E }
In graph G K every node will be given a power-function v i ( t ) ; for node V i and node V j , if edge E i j ∈ E , then on the edge E i j exists a power-function
e i j ( t ) = v j ( t ) − v i ( t ) (3)
In Formulation (3), lower reaches node is left and upper reaches node is right, in opposite direction would sign a negative in the front of e i j ( t ) .
In graph G K if the edge E i j exists, then give a flow-function f i j ( t ) , its direction is from the upper reaches node to the lower reaches node. The graph G K would be satisfied by 10.2., 10.3., 10.4., 10.5. as below.
In the graph G K every cycle W i j is satisfied by
∑ if E i j ∈ W i j e i j ( t ) = 0
In the graph G K every node V i is satisfied by
∑ if E i ∈ V i f i ( t ) = 0
as above input flow f i ( t ) is positive and output flow f i ( t ) is negative.
Defines
R i j ( t ) = e i j ( t ) f i j ( t )
is the contradiction-function of dge E i j in graph G K .
In the graph G K every edge E i j , make
W i j = ∫ t 0 t 1 e i j ( t ) f i j ( t ) d t
W i j is called what work done in [ t 0 , t 1 ] by contradiction-function R i j ( t ) on edge E i j .
1) Graph G K have n couples of nodes and two nodes V + 1 , V − 1 , total 2 ( n + 1 ) ;
2) Graph G K is a no-loop, no-multiple edge directed simple graph;
3) In the graph G K , n + 1 positive nodes construct a perfect subgraph, another n + 1 negative node construct another perfect subgraph;
4) Node V + 1 connect only to node V − 1 , positive node V q + connect only to negative node V q − what right subscript is equal.
5) The power-function of node V + 1 is a constant +1，the power-function of V − 1 is a constant −1, the power-function of edge E + 1 , − 1 is a constant +2, the flow-function of edge E + 1 , − 1 is a constant I;
6) In the graph G K , other nodes and edges all be defined power-function v i ( t ) , power-function e i j ( t ) , flow-function f i j ( t ) and contradiction-function R i j ( t ) , these are satisfied by Kirchhoff laws as above.
1) For t ∈ [ t 0 , t 1 ] , if the 1-Order derivative of function Q i j ( t ) exist and if
d Q i j ( t ) d t ≠ 0
then Q i j ( t ) is called variation, or not is called heredity.
2) Every contradiction-function R i j ( t ) always can be seen an algebra-sum of a constant C and function R i j ( t ) i.e.
R i j ( t ) = Q i j ( t ) + C (4)
and if
d Q i j ( t ) d t ≠ 0
then in Formulation (4) Q i j ( t ) is variation and C is heredity.
3) Heredity and variation theorem
Heredity and variation is basic law in object-logic of cause in subject-logic.
Prove: combining the 11.1. and 11.2. as above which can be proved obviously.
For the contradiction-function R i j ( t ) , if its m-order derivative exist and the points as below make
d k R i j ( t ) d t k = 0 , or ∞ , k = 0 , 1 , ⋯ , m ;
and some special-selected points are called critical-point.
Property-Function Critical-Point TheoremThe existence of critical-points will make what some new property-function is born or some old property-function is dead.
If the rank-number of logic-variable A i ± s ( t ) , A j ± r ( t ) is equal i.e. s = r , then A i ± s ( t ) and A j ± r ( t ) is called isomorphic-equality, denote isomorphic-equality as
A i ± s ( t ) ↔ A j ± r ( t )
Isomorphic-equality ↔ is satisfied by as below reflexive law A j ± r ( t ) ↔ A j ± r ( t ) , A i ± s ( t ) ↔ A i ± s ( t ) symmetrical law if A j ± r ( t ) ↔ A i ± s , then A i ± s ( t ) ↔ A j ± r ( t ) transtive law if A i ± s ( t ) ↔ A j ± r ( t ) and A j ± r ( t ) ↔ A m ± k ( t ) , then A i ± s ( t ) ↔ A m ± k ( t )
1) Definition: the two logic-variable A i ± s ( t ) and A j ± r ( t ) is called similarity, if
a. A i ± S ( t ) ↔ A j ± r ( t ) ;
b. in the graph G K i and G K j , their contradiction-function of corresponding edge is proportion i.e.
R i j i ( t ) = C R i j j ( t ) , R i j i ( t ) ∈ G K i , R i j j ( t ) ∈ G K j
in formulation as above C is a constant.
2) If A i ± s ( t ) and A j ± r ( t ) is called similarity, what is denoted as A i ± s ( t ) ~ A j ± r ( t ) . The relation is satisfied as below reflexive law A i ± s ( t ) ~ A i ± s ( t ) , A j ± r ( t ) ~ A j ± r ( t ) symmetrical law if A i ± s ( t ) ~ A j ± r ( t ) , then A j ± r ( t ) ~ A i ± s ( t ) transtive law if A i ± s ( t ) ~ A j ± r ( t ) , and A j ± r ( t ) ~ A m ± k ( t ) , then A i ± s ( t ) ~ A m ± k ( t )
3) Thinking analogy theorem
Two of logic?variable A i ± s ( t ) and A j ± r ( t ) are analogy-able in thinking, if and only if
A i ± s ( t ) ~ A j ± r ( t )
Proof: based on the definition of similarity, if
A i ± s ( t ) ~ A j ± r ( t )
then
A i ± s ( t ) = C A j ± r ( t )
In formulation as above, C is a constant, thus they are analogy-able in thinking.
Proof is Over.
As shown above, author has established an axiom system depending on several laws, some definitions, graph GK and Mozi-principle, and proved some theorems for dialectical logic K-model. The advanced properties and theorems for dialectical logic K-model will be explained in succedent papers by the author.
Jiang, Y.Z. (2017) Axiom System and Some Theorems for Dialectical-Logic K-Model. Open Access Library Journal, 4: e3817. https://doi.org/10.4236/oalib.1103817