Objective primary algebra

Objective Brownian Algebra PICTORIAL SYMBOLIZATION OF BOOLEAN ALGEBRA

FOREWORD FOREWORD In 1970-s George Spencer-Brown wrote the controversial book titled The Laws of Form.

It is controversial because he used the VOID in the algebraic equations in his new algebra. The other strange symbol he used is the half box he called the CROSS. With the new symbols he constructed an arithmetic based on the Law of Cancelation and the Law of Condensation.

Based on the arithmetic he developed a Primary Algebra with two primitives: the law of Position and the Law of transposition.

In the later part of his book he interpreted his primary algebra as a new planar symbolization of Boolean algebra.

Next, Louis Kauffman replaced the CROSS with the complete BOX to develop his semi pictorial Box Algebra as the symbolization of Boolean algebra.

In this book we replace the letters in the Box Algebra with colored object to construct a totally pictorial planar symbolization of Boolean Algebra called Objective Brownian Algebra.

PRIMITIVES of the OBJECTIVE BROWNIAN ALGEBRA The Objective Brownian Algebra, as

the symbolization of the Boolean Agebra of Logic, is based on two primitive concepts:

VOID BOX

represents FALSE and represents TRUE.

The primitive operations are Juxtaposition represents a OR b BOX enclosure represents NOT a The primitives of logical arithmetic

are

Cancelation Condensation

The primitives of logical algebra are Position

Transposition

Primary Objective Algebra Initials Position Transposition

Considering AND is equal to NOT

((NOT a) OR (NOT b)),

the position equation is nothing but the law of contradiction and the transposition equation is nothing but the law of distribution of OR over AND

Primary algebra Rules of Inference Using the algebraic rules

and

we can derive all the boolean tautologies, from the position and transposition axioms, as consequences.

CONSEQUENCES Some of the simple Boolean

tautologies that can be proved as consequences of objective Brownian algebra are

Reflexion

Generation

Complementation

Integration

Occlusion

Iteration

Extension

Proof of Reflexion

Proof of Generation

Proof of Complementation

Proof of Integration

Proof of Occlusion

Proof of Iteration

Proof of Extension

AFTERWORD Brownian algebra is not the simplest axiom

system for the Boolean algebra of logic.

In fact, if we take the Occlusion, the Reflexion and the Generation laws as axioms to build the Boundary Logic of William Bricken as another simple axiomatic formulation of the Boolean Algebra Ultimately, we can simply use the Extension law as the only axiom for the Box Algebra as the simplest axiomatization of the Boolean Algebra All of them can be made Objective by using colored objects as variable names, the box as NOT operator and juxtaposition as OR connection.

Objective Algebra can also represent logical system based on AND and NOT operation such as Peircean Existential Graph System.

References Aristotle :

Non-Mathematical Verbal Logic

http://classics.mit.edu/Aristotle/prior.1.i.html

George Boole:

Algebraic Symbolic Logic (Algebra of Logic) http://www.freeinfosociety.com/media/pdf/4708.pdf

Charles Sanders Peirce:

Algebraic Graphical Logic (Existential Graph) http://www.jfsowa.com/peirce/ms514.htm

George Spencer-Brown:

Algebraic Graphical Logic (Laws of Form)

http://www.4shared.com/document/bBAP7ovO/G-spencer-Brown-Laws-of-Form1.html

Louis Kauffman:

Algebraic Pictorial Logic (Box Algebra)

http://www.math.uic.edu/~kauffman/Arithmetic.htm