The simple existential graph system

SIMPLE EXISTENTIAL

GRAPH SYSTEM A R M A H E D I M A H Z A R © 2 0 1 3

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.

S I M P L E E X I S T E N T I A L G R A P H S Y S T E M 2

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.


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


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


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


IMPLICATION

IMPLICATION

EQUIVALENCE


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.

.


THEOREM P5 DOUBLE CUT

Proof of

N0 consistency

commutation

D1 definition of ->

Proof of

N0 concistency”

N0 consistency

P4 deiteration

D1 definition of ->


LEMMA 1: INVERSION

Proof

consistency

definition

double cut

definition of ->


THEOREM P0

VOID

Proof

consistency

= definition

= substitution

= VOID double cut


LEMMA 2: ADDITION

Proof

consistency

definition

iteration

double cut

definition


LEMMA:

0-DEPTH DELETION

proof

consistency

definition iteration definition

Lemma: 1-DEPTH INSERTION Proof 0-depth deletion inversion


LEMMA:

2-DEPTH DELETION

proof

1-depth insertion

addition

inversion


LEMMA: 3-DEPTH INSERTION proof

2-depth

deletion

addition

inversion

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


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.


The simple existential graph system

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