Colorful Bar Algebra

Step Two to Games of Thing:
Colorful Bar Algebra

In my last Blog,  I mentioned that I discovered different forms of pictorial algebra of logic.

In this letter I will explain one of such form algebra: the Bar Algebra.

One of the pictorial algebra is the Box Algebra of Louis Kauffman. If we eliminate the left and bottom edges of the boxes in all valid logical formulas, then we will get all form equation of the Cross Algebra of

George Spencer-Brown.  So actually, the Cross Algebra is the half, pictorially speaking, of the Box Algebra. It seems that Cross Algebra is nothing but a simplification of the Box Algebra. In this letter, I will show you that Cross algebra can be more simplified. This is what I have done.

Let us  we erased the horizontal legs of the Brownian Crosses in any formula, then we will get formulas with just overbars. I will call such pictorial algebra as the Bar Algebra. The Bar algebra seems to be the quarter of box algebra.  The following is the description of the Bar Algebra

Bar Arithmetics

As in the Box Arithmetics, the logical Constant FALSE is represented by

VOID

and TRUE is drawn as an overbar

__

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

NOT a is drawn as a horizontal bar above a

a OR b is drawn as a beside b

NOT (a OR b) is drawn as a bar above a beside b

The algebra is based on the Bar Arithmetic. The logical primitives of logical Bar arithmetic

is the Law of Negation (or Cancelation) and the Law of Disjunction (or Condensation).

Law of Cancelation [ [ ] ] =  is drawn as

Law of Condensation [ ] [ ] = [ ]  is drawn as

Algebra of Bars

As we know, George Spencer-Brown discovered that the whole Boolean algebra can be based on just two logical identities which he called as position and 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 Bar Algebra

The 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 just  a theorem in the original Brownian Cross Algebra.

The Huntington Axiom [[a][b]][[a]b]=a in Brownian Cross Algebra can be drawn as

  

Last remarks

Well, somebody, who is well studied the history of symbolic logic, will recognize the the formulas of the Bar Algebra using overbars is nothing but the boolean algebra written with OVERBAR as the symbol of negation and with NOTHING replacing the symbol + and 0 in it. So actually it is not new after all. The newness is just in the notation, where the use of NOTHING as symbols that will make some people perceive it as ridiculous.  Now, I have to admit my discovery is nothing new, but in the next blog I will show everybody that if we replace over-Bar with under-V we will get a more beautiful V Algebra of Logic, hopefully.