PICTORIAL SYMBOLIZATION OF BOOLEAN ALGEBRA

Objective Brownian Algebra

FOREWORDFOREWORD

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 represents FALSE and

� BOX 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 AlgebraInitials

� 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 algebraRules 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 ofReflexion

Proof ofGeneration

Proof ofComplementation

Proof ofIntegration

Proof ofOcclusion

Proof ofIteration

Proof ofExtension

AFTERWORD

� Brownian algebra is not the simplest axiomsystem for the Boolean algebra of logic.

� In fact, if we take the Occlusion, theReflexion and the Generation laws asaxioms to build the Boundary Logic ofWilliam Bricken as another simpleaxiomatic formulation of the BooleanAlgebra

� Ultimately, we can simply use theExtension law as the only axiom for the BoxAlgebra as the simplest axiomatization ofthe Boolean Algebra

� All of them can be made Objective by usingcolored objects as variable names, the boxas NOT operator and juxtaposition as ORconnection.

� Objective Algebra can also represent logicalsystem based on AND and NOT operationsuch 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-Form-1.html

� Louis Kauffman: � Algebraic Pictorial Logic (Box Algebra)

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