Algebra of Forms

Algebra of Forms
Armahedi Mahzar (c) 2015

George Spencer-Brown   in the 70s of the last century write a strange book titled ‘Laws of Form’. In this book he used CROSS   as a strange pictorial notation where NOT x is drawn as

 

and he also used EMPTY character as FALSE. So, it is hard to be typed into the screen using ASCII characters.

Fortunately, William Bricken

  make it easier to type them using parentheses [  ] as symbolic notation for NOT. NOT x is written as [x]. With this notation all strange pictorial forms of Brown can be written as strings of characters which is easy to write with computer keyboards and easy to read in computer monitors.

In this new notation, the two Brownian algebra primitives are

J1: [[x]x]=                         (the law of position)

and
J2: [[xz][yz]]=[[x][y]]z      (the law of transposition).


I was glad to read that logic now only based on just two axioms.
However, to my surprise, the axiomatization of logic is not unique. William Bricken used the first three consequences of Brownian primary algebra,

C1: [[x]] = x                  (Law of reflection),
C2: [xy]y = [x]y            (Law of generation) and
C3:  [ ]x = [ ]                 (Law of integration)
as an alternative axiom-base.

Louis Kauffman   used the sixth consequence

C6: [[x]y][[x][y]] = x     (the law of extension)


as the single axiom to base the whole Boolean algebra.
Kauffmanian Box Algebra is the simplest axiomatization thay I have read.

In my previous blog, we see that the arithmologic of Fred Sommers  can also derive all Boolean tautologies. Its simplest theorem is x->x (Law of consistency) or -x+x=0 which can be written in Bricken notation as [x]x=   if we use Sommersian interpretation x+y as x OR/AND y and VOID as TRUE/FALSE.

Luckily, this Law can be made as the single axiom for the algebra of forms. This statement will be shown in this blog entry:

Sommersian algebra

The single axiom of Sommersian algebra is

   [x]x =     (Law of Consistency)

From this axiom we can derive three lemmas: Double Negation, De Morgan Law and Boole Index Law.

Lemma 1: Law of Double Negation
Deductive proof:
[[x]]
=[[x]][x]x        {by consistency axiom}
=        x            {by consistency axiom} QED

Lemma 2: De Morgan Law [xy]=[x][y]
Deductive proof:
[xy]
= [x]x[y]y[xy]  {by Consistency axiom}
= [x][y][xy]xy  {by implicit commutation}
= [x][y]             {by consistency axiom} QED

Lemma 3: Boole index law
Deductive proof:
[y]y=                                        {consistency axiom}
[xx]xx=                                    {substitution y=xx}
[xx]xx[x]x=                             {substitution VOID=[x]x}
[xx]x[x]xx=                             {implicit commutation}
(xx->x) AND (x->xx)=TRUE {definition of ->}
(xx=x) is TRUE                      {definition of = } QED

Kauffman Agebra

With the aid of the three lemmas we can prove
Kauffman axiom as a theorem of Sommersian algebra

Deductive proof:
[[a]b][[a][b]]
= [[a]][b][[a]][[b]]  {Lemma 2: De Morgan Law}
=   a  [b]  a    b        {Lemma 1: Law of double Negation}
=   a       a                {Consistency axiom}
=   a                         {Lemma 3: Law of Index} QED

Consequently, all consequences of Kauffmanian algebra, which are identical to all Boolean tautologies, are also theorems for the Sommersian primary algebra.

Brownian Primary Algebra

Four of the theorems of Sommersian algebra are the primitives of the Brownian primary algebra.
In the following are the proofs of Brownian primitives as theorems.

Proof of the Law of Cancellation [[ ]]=

[[x]]=x                        {lemma 1: law of double negation}
[[ ]]=                           {substitution x=   } QED

Proof of the Law of Condensation [ ][ ]=[ ]
xx = x                         {lemma 3: index law}
[ ][ ]=[ ]                      {substitution x=[]} QED

Proof of the Law of Position [[x]x]= 
[[x]x]      
=[[ ]]                           {substitution x= }
=                                 {lemma 1: law of double negation} QED

Proof of the Law of Transposition
[[xz][yz]]             
=[[x][z][y][z]]            {Lemma 2: De Morgan Law}
=[[x][y]][[z]][[z]]       {Lemma 2: De Morgan Law}
=[[x][y]]  z    z            {Lemma 1: Double negation law}
=[[x][y]]  z                  {Lemma 3: Index Law} QED

Afternotes

1. I think Sommersian Algebra is the simplest algebra of forms since its dingle axiom is simpler than the Kauffman axiom.

2. The most inconventional notation is that Sommersian interpretation is the mixing Brownian interpretation and Peircean interpretation. So, in the De Morgan Law equation the left side is read as Brownian and the right side is read as Peircean or the other way around. It is also true for the reading of consistency axiom.

3. Actually, we can transform the Sommersian algebra of logic into a logic game of things that is simpler than the Boolean logic game or the Peircean one. I will try to show it in my next blog entries.

4. Of course I am still opening my mind for any criticism, so I can correct my perception on the ultimate simplicity of logic. Hopefully, some readers of this post will correct my wrong perception, so I will be more enlightened.