3. While other systems like boundary logic provide alternative axi

1. Objective Brownian
Algebra
PICTORIAL SYMBOLIZATION OF
BOOLEAN ALGEBRA

2. 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.

3. 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
BOX enclosure
represents a OR b
represents NOT a
The primitives of logical arithmetic
are
Cancelation
Condensation
The primitives of logical algebra are
Position
Transposition

4. 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

5. 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.

6. 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

7. Proof of
Reflexion

8. Proof of
Generation

9. Proof of
Complementation

10. Proof of
Integration

11. Proof of
Occlusion

12. Proof of
Iteration

13. Proof of
Extension

14. 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.

15. 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