The document presents a simplified existential graph system based on the original system developed by Charles Sanders Peirce. The simplified system has one axiom ("consistency") and one rule of inference ("iteration"), making it simpler than Peirce's system which had one axiom and five rules of inference. The author proves that the rules and axiom of Peirce's system can all be derived as theorems in the simplified system, showing their logical equivalence. This establishes the simplified existential graph system as a valid axiomatization of Boolean algebra.

1. SIMPLE EXISTENTIAL
GRAPH SYSTEM
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2. INTRODUCTION
In the nineteenth century, the english man George Boole made a
revolutionary step in logic by using mathematical symbols and
method to study it. X OR Y, X AND Y and NOT X are
symbolized by X+Y, X x Y and 1-X respectively
The next revolutionary step was made by the american Charles
Sanders Peirce by replacing the linear mathematical with the
planar pictorial symbols and operations. X AND Y is pictured as
X Y and NOT X is pictured by X enclosed by an oval.
The pictures are called as existential graphs and implication is
represented by rules of inference. All boolean identities can be
created from empty sheet representing TRUE.
I have discovered that if we replace TRUE the Law of
Consistency, then we get a new simple existential system.
In the following pages I will prove that the new simple existential
graph system is logically equivalent to the original existential
graph system of Peirce.
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3. PEIRCE RULES OF INFERENCE
Rl. The rule of erasure.
Any evenly enclosed graph may be erased.
R2. The rule of insertion.
Any graph may be scribed on any oddly enclosed
area.
R3. The rule of iteration.
If a graph P occurs on SA or in a nest of cuts, it
may be scribed on any area not part of P, which is
contained by {, }.
R4. The rule of deiteration.
Any graph whose occurrence could be the result
of iteration may be erased.
RS. The rule of the double cut.
The double cut may be inserted around or
removed (where it occurs) from any graph on any
area.
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4. OBJECT LOGIC
In this book, we will use boxes as enclosures
and colored marbles as variables in
an algebraic system called Object Logic.
Object logic is nothing but the fully pictorial
representation of the Box Algebra discovered by
Louis Kauffman.
In the Object Logic,
 TRUE is represented by VOID
 AND is represented by JUXTAPOSITION
 NOT is represented by BOX enclosement
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5. EXISTENTIAL GRAPH SYSTEM
IN OBJECT LOGIC
The existential Graph System of Peirce has only one axiom
P0 Truth : VOID
and five rules of inference
P1 even deletion:
when is an even number of nested
P2 odd insertion: <x>-><gx>
when <..> is an odd number of nested
P3 iteration: g[x]->g[gx]
P4 deiteration: g[gx]->g[x]
P5 double cut: [[x]]=x
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6. SIMPLIFYING
EXISTENTIAL GRAPH SYSTEM
The original Existential Graph System of Charles Sanders
Peirce is the most simple axiomatization of Boolean algebra
since it has only one axiom which is nothing but TRUE which
is represented by VOID.
Unfortunately, it has five rules inference, so in fact it is in the
same complexity to the propositional calculus in the book
Principia Mathematica written by Alfred North Whitehead and
Bertrand Russel which has five axiom and single inference
rule.
That’s why I try to simplify the peircean Existential Graph
System and I am glad to find out that if we replace the TRUE
axiom with the Law of CONSISTENCY, then we only need
single inference rule: the ITERATION
So finally I discovered the Simple Graph System
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7. IMPLICATION
IMPLICATION
EQUIVALENCE
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8. SIMPLE EXISTENTIAL GRAPH SYSTEM
My system has only one axiom:
M0 : CONSISTENCY
and just one rule of inference:
M1: ITERATION
so it is simpler than the Peircean existential graph system .
In the following pages, I will prove that all inference rules, P1, P2,
P3, P4 & P5, and the single axiom P0 in the peircean Existential
Graph System is nothing but theorems of the new simple
Existential Graph System.
.
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9. THEOREM P5 DOUBLE CUT
Proof of
N0 consistency
commutation
D1 definition of ->
Proof of
N0 concistency”
N0 consistency
P4 deiteration
D1 definition of ->
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10. LEMMA 1: INVERSION
Proof
consistency
definition
double cut
definition of ->
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11. THEOREM P0
VOID
Proof
consistency
= definition
= substitution
= VOID double cut
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12. LEMMA 2: ADDITION
Proof
consistency
definition
iteration
double cut
definition
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13. LEMMA:
0-DEPTH DELETION
proof
consistency
definition
iteration
definition
Lemma:
1-DEPTH INSERTION
Proof
0-depth deletion
inversion
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14. LEMMA:
2-DEPTH DELETION
proof
1-depth
insertion
addition
inversion
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LEMMA:
3-DEPTH INSERTION
proof
2-depth
deletion
addition
inversion

15. THEOREM P1: EVEN-DEPTH DELETION
2n-depth deletion can be proved by inverting the
addition of the (2n-1)-depth insertion
Theorem P2: ODD-DEPTH INSERTION
(2n+1)-depth insertion can be proved by inverting
the addition of the 2n-depth deletion
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16. AFTERWORD
Since the new existential graph system is the generator of all
Boolean identities., it is isomorphic to Boolean algebra as the
propositional calculus is. There are equational systems that are
also isomorphic to Boolean algebra such as the Brownian cross
algebra, the Kaufmanian Box Algebra and the Brickenian
Boundary Logic algebra.
Brickenian boundary logic algebra has three axioms (dominion,
pervasion and involution) and the simple existential graph system
has two primitive ( consistency axiom and ieration rule), so the
later is simpler than the former. It is simpler because it use the
implicational rules of inference, so the involution or double cut
can be derived as a theorem.
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