Colorful V Algebra

Step Three to Games of Thing:
Colorful V Algebra

In my last blog I expounded the possibility of formulating the logic algebra in a pictorial symbols which I call as Bar Algebra. Where the bar is the horizontal line segment that is arranged vertically and horizontally. This Bar Algebra is obviously isomorph to Brownian Cross Algebra. But the over-bar can be replaced with under-V to get the following colorful V Algebra

V Algebra

Boolean algebra can be formulated in 2-dimensional manner as the Box Algebra of Kauffman or the Cross Algebra of Spencer-Brown. Boolean algebra can also be 2-dimensionally reformulated as tree algebra.

The logical Constants FALSE is represented by VOID

and TRUE is drawn as V

  

We can also represented the NOT and OR operation as drawings in the V Algebra.

NOT a is drawn as

or  ,

a OR b is drawn as

V Arithmetic

The algebra is based on the V arithmetic. The logical primitives of the logical V arithmetic is the Law of Negation or Cancelation and the Law of Disjunction or Condensation as it is called by George Spencer-Brown.

In the pictorial V arithmetic,  the Law of Cancelation [ [ ] ] =   can be drawn as

The Law of Condensation [ ] [ ] = [ ] can be drawn as

Algebra of Vs

The Brownian algebra based on the primary arithmetics can also be reformulated using colored V. The Brownian algebra has only two axioms: the Law of Position and the Law of Transposition.

Law of position  [[a]a] =  is drawn as

The Law of Distribution or the Law of Transposition [[ac][bc]]=[[a][b]]c can be drawn as

Simplifying V algebra

he Brownian Algenra is a very simple description of Boolean algebra by using just two axioms. However Louis Kauffman simplified it to a Box algebra with just one axiom the Huntington Axiom [[a][b]][[a]b]=a which is in fact a theorem in the original Brownian Cross Algebra.

The single axiom of Kauffmanian Box Algebra can be drawn as 

Last remarks

I like the colorful V pictorial symbolism better the box algebra, but watching the Vs more deeply,

we will meet the difficulties to represent the string of OR expression such as (a + b + c)’ . To overcome such difficulties, I found out another pictorial symbolic formulation of Boolean algebra which I call as T Algebra.

My next blog will discuss the T Algebra.