The simple existential graph system

SIMPLE EXISTENTIAL GRAPH SYSTEM ARMAHEDI MAHZAR © 2013

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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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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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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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 SIMPLE EXISTENTIAL GRAPH SYSTEM

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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.

Thatâ&#x20AC;&#x2122;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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IMPLICATION IMPLICATION

EQUIVALENCE

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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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THEOREM P5 DOUBLE CUT

Proof of N0 consistency commutation

D1 definition of -> Proof of N0 concistencyâ&#x20AC;?

N0 consistency P4 deiteration

D1 definition of ->

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LEMMA 1: INVERSION

Proof consistency

definition

double cut

definition of ->

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THEOREM P0 VOID

Proof

consistency =

definition

=

substitution

=

VOID

double cut

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LEMMA 2: ADDITION

Proof consistency

definition iteration

double cut

definition

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LEMMA: 0-DEPTH DELETION proof consistency definition iteration

definition

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

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LEMMA: 2-DEPTH DELETION proof

1-depth insertion addition

inversion

LEMMA: 3-DEPTH INSERTION proof

2-depth deletion addition

inversion

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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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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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