An emergence of formal logic induced by an internal agent

Koji Sawa The Senior High School, Japan Women’s University, Japan

Yukio-Pegio GunjiKobe University, Japan

FIS2010Beijing, China, Aug 21-24, 2010

Proposal

• A dynamical model of formal logic

– It is autonomously transformed.

– It is composed of a system and its subsystem.

– It is represented as transformation of directed graphs.

Motivations 1: Logic

• Where does logic come from?

• Our previous work:Dialogue models as the origin of logic (Sawa and Gunji, 2007, 2008)

– Each model is represented in the form of a multi-agent model.

Motivations 2: Multi-agent model

• The behavior of a system is influenced by agents and interactions between agents.

→ System is not autonomous.

• Agent– autonomy, sociality, ...

→ Agent is external to system.

A connection with FIS• Brenner (2010). Information in Reality. Logic and Metaphysics

“every real complex process is accompanied, logically and functionally, by its opposite or contradiction (Principle of Dynamic Opposition), but only in the sense that when one element is (predominantly) present or actualized, the other is (predominantly) absent or potentialized, alternately and reciprocally, without either ever going to zero”

→ We realize a concept touching on above by the invalidation of reflexive law.

• Hofkirchner (2010). Four ways of thinking in information “Reductionism, Projectivism, Disjunctivism, and Integrativism”

→ In my opinion, Reductionism and Projectivism correspond to deduction and induction, respectively. Just as Hofkirchner claims that Integrativism must be needed, so we also consider that the third inference abduction must be needed (cf. Sawa and Gunji, in press).– Actually in this presentation, we do not treat these inferences directly, however these

inferences are in the scope of our study.

A connection with FIS• Collier (2010). Kinds of Information in Scientific Use

“For each kind of substantive information used in the sciences there is a distinct level formed by bifurcations that form cohesive structures at the next higher level. This is reflected in the information at each level, which inherits the properties of the lower level, but produces new asymmetries at its own level through the formation of new cohesions peculiar to the level.”

→ We propose an idea of the way to raise a level presented above: a representation by nonhierarchical, divisible, and incorporable objects.

Model

Multi-agent model

• Each agent is autonomous.　→　 Agent is independent and external to

system.　→　 System refers external.

System

Agent

Interaction

“Emergence” Restriction

Internal Agent Model

• Internal agent := A part of a system.– Internal agent is sometimes abbreviated to agent.

• System never refers external.– Internal measurement (Matsuno, 1989)

• S-IA interaction := Interaction between system and internal agent.

System

Agent

Interaction

“Emergence” Restriction

Formal logic represented by a directed graph

Directed Graph

Object

Arrow

Object

Implicational relation

Identity and obviousness of object

• A implies A.– A is A.– There is no doubt about the obviousness of

object.

• Derivation of LK

A A├ B B├, A A B B ├ C C├

, , A A B B C C ├, A B B C A C ├

Assuming the obviousness of object

Soft object

• Soft object := a cycle of arrows

• Example

XSoft Object

Soft object

• Identity: X → X

• IfX → Y, Y → Z, Z → X,

thenX XY Y Z Z.(assuming transitive law)

X

ZY

Soft Object

Soft object

• Soft object := a cycle of arrows

• Example

Soft(breakable)

Hard（ nonbreakable ）

Number of arrows

less more

Identity and obviousness of object

• Equivalence law: (Condition that a set is treated as one unit)– Reflexive law: A → A– Symmetric law: A → B implies B → A– Transitive law:

A → B and B → C implies A → C• A soft object (except the hardest one (a

complete graph)) is an object in which the equivalence law is partially invalidated.

Soft arrow

• Soft arrow :=a bundle of arrows in the same direction.

• Example

Soft(Breakable)

Hard(Nonbreakable)

Number of arrows

Less More

Summary of model from a logical perspective

• Formal logic– Represented by a directed graph.– Consists of objects and arrows.

• Object– Represented by a cycle of arrows.– Soft object

• Arrow– Represented by a bundle of arrows– Soft arrow

×

×

• Agent influences system through pursuit of agent’s “purpose”.

• System influences agent through pursuit of system’s “purpose”.

Interaction between system and agent in formal logic

System

Agent

Transitivity Rate (TR)

• Def. Given a directed graph G,

where : the number of arrows in G, : the graph transformed from

G, in which the transitive law

holds completely by adding

requisite arrows.

TR : | | / | | ,G G

| |GG

Transitivity Rate (TR)

TR=3/4=0.75

• Transitivity rate (TR) is one of measures of reliability of a directed graph as formal logic.

• Agent’s purpose := increase of TR.

• Example

Assumingtransitive law

S-IA interaction　 Agent → System

– Add an arrow satisfying below conditions to system

• increases TR of agent;• does not exist in system;• shares at least one node with

arrows of agent.

　 System → Agent

– Add an arrow satisfying below conditions to agent

• increases TR of system;• does not exist in agent;• shares at least one node with arrows of agent.

System

Agent

S-IA interaction:succession of applications of transitive law to two parts: system and agent.

Example of time transitions by S-IA Interaction

System

Agent

t = k

t = k + 1

Random graph ?

S-IA Interaction

Trial 1

• What kind of graphs emerge by S-IA interaction?

Result of Trial 1

Compress

• Initial random graph (50 nodes)– All arrows:

System– A subset of arrows: Agent

• Convergent graph– There are soft objects and soft

arrows among soft objects.– All soft objects and soft arrows are

hardest ones.– Transitive law holds among soft

arrows.

• In sum, a graph representing formal logic in which the transitive law holds completely.Another result

• Trial 2What happens if the obviousness of objects is invalidated in the emergent graph representing formal logic?

Invalidation of the obviousness of objects= Invalidation of reflexive law (A → A)= Elimination of arrows in soft objects

Random graphGraph representing

formal logic

S-IA Interaction

Trial 2• Trial 1

Initial random graph of Trial 20 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 10 0 0 0 0

0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

• Invalidation of the obviousness of objects

• Softening of arrows

Choose arrowsRate: q

Hardest soft arrows

System

Agent

Choose arrowsRate: p

S-IA Interaction

Initial graph (p=1.0, q=0.75)

System (100 arrows) Agent (82 arrows)

0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Result 1 (p=1.0, q=0.75)• Soft objects and soft arrows

emerge as the hardest ones.• Transitive law holds among

soft arrows.

• Convergent graph represents formal logic.

11

12

13

14

15

6

7

8

9

10

21

22

23

24

25

1

2

3

4

5

16

17

18

19

20

0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0

Result 2 (p=0.5, q=0.5)

11

12

13

14

15

6

7

8

9

10

21

22

23

24

25

1

2

3

4

5

16

17

18

19

20

0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 1 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 10 0 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 0 0 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 1 1 10 0 0 0 1 1 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 1 1 1 1 10 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 10 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 1 1 1 10 0 0 0 1 1 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 10 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 1 0 1 10 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0

Summary of results of Trial 2

• Convergent graph represents formal logic.– Soft objects and soft arrows emerge as the hardest ones.– Transitive law holds among soft arrows.

• “Latent” objects expected from soft arrows become valid objects.– Emergence of definite (=valid=“hardest”) concept– Furthermore, emergence in the different forms than expected

ones

• Internal Agent Model realizes dynamical formal logic,– in which logical structure is roughly retained.

Summary of results of Trial 2

X Y

A∧B A∨B

A

B

X = A∧B Y = A∨B

Summary of results of Trial 2

(A) Number of soft objects consisting of multiple nodes

(B) Number of singletons (soft objects consisting of only one node)

(C) = (A) + (B)

(D) Number of soft objects which are composed of nodes of different latent objects

qp (A) (B) (C) (D) (D)/(C) (A) (B) (C) (D) (D)/(C) (A) (B) (C) (D) (D)/(C) (A) (B) (C) (D) (D)/(C)1.00 (1) 9 0 9 1 0.11 8 4 12 0 0.00 9 7 16 2 0.13 5 0 5 4 0.80

(2) 8 1 9 2 0.22 6 4 10 0 0.00 6 0 6 3 0.50 4 2 6 3 0.50(3) 8 0 8 0 0.00 8 1 9 2 0.22 6 1 7 3 0.43 4 0 4 3 0.75

Ave. 0.11 0.07 0.35 0.680.75 (1) 7 0 7 2 0.29 8 3 11 5 0.45 6 2 8 3 0.38 5 0 5 4 0.80

(2) 9 3 12 3 0.25 9 3 12 5 0.42 6 0 6 4 0.67 5 0 5 4 0.80(3) 8 2 10 4 0.40 8 2 10 3 0.30 2 0 2 1 0.50 4 0 4 2 0.50

Ave. 0.31 0.39 0.51 0.700.50 (1) 11 1 12 4 0.33 6 5 11 5 0.45 10 3 13 5 0.38 6 1 7 4 0.57

(2) 7 9 16 1 0.06 11 0 11 4 0.36 5 2 7 4 0.57 3 1 4 3 0.75(3) 9 2 11 5 0.45 7 4 11 3 0.27 6 5 11 4 0.36 3 3 6 2 0.33

Ave. 0.28 0.36 0.44 0.550.25 (1) 9 5 14 4 0.29 8 1 9 8 0.89 5 3 8 4 0.50 8 0 8 6 0.75

(2) 8 4 12 7 0.58 11 2 13 6 0.46 9 2 11 6 0.55 7 1 8 6 0.75(3) 10 1 11 7 0.64 8 5 13 6 0.46 9 5 14 9 0.64 9 1 10 8 0.80

Ave. 0.50 0.60 0.56 0.77

1.00 0.75 0.50 0.25

q 1 0

Agent

p 1

0

Similar toformer logic

Dissimilar toformer logic

System

Discussions

Discussion 1:From a logical perspective

• Premise– Reflexive law (A → A) is invalidated.

• This corresponds to invalidation of the obviousness of the object.

– Transitive law (A → B and B → C implies A → C) is treated as S-IA interaction,

• which is succession of applications of transitive law to system (whole) and agent (part).

• Result– Emergence of objects (Trial 1),

• as the hardest ones.• Arrows also emerge as hardest ones.

– Emergence of objects expected from arrows (Trial 2),• in the different forms than expected ones.• This emergence corresponds to revision of objects due to relations

(arrows) of objects.

Discussion 2: Object and agent

• In Internal Agent Model, both soft object and internal agent are mere subgraphs of system.

• Soft object– is an alternative to an ordinary object:

• nonhierarchical, • divisible, • incorporable.

– represents a concept.– takes on a spatial extent.

• Internal agent– is an object which has purpose.

• In Internal Agent Model, internal agent purposes the adequacy of the system as formal logic.

– takes on a temporal extent.

Future studies

• Internal Agent ModelAgent (purpose) → Soft object (concept).

• We would like to treatSoft object (concept) → Agent (purpose),– by the argument of the positional relation or inclusive relation

among soft objects.

• Mediation of Object-Relation Model(Sawa and Gunji, in press)– represents expansion and contraction of objects and relations

among objects.– This model implies two fundamental logical inferences, deduction

and induction in the form of classification of C. S. Peirce. In addition, it also implies the third inference of Peirce, abduction, which is usually disregarded.

Thank you very muchfor your attention.