ARITHMOLOGIC 

Armahedi Mahzar (c) 2015

George Spencer-Brown  reduced logic algebra to an axiomatic system based on just two axioms (contradiction and disteribution) with algebraic inference rules.

Louis Kauffman  simplified the Brownian system into a Box Algebra based on single axiom (reductio ad absurdum) also with the same algebraic inference rules. The axiom actually is one of the three Huntington axioms. The other two axioms are not necessary in a pictorial symbolization.

Charles Sanders Peirce reduced propositional calculus to an existential graph system based on the simplest axiom: TRUE. However he used 5 logical inference rules. In fact, Peirce’s system is not a simple system at all, Regarding rules of inferences as primitives proposition accompanying the single axiom, we can conclude that the system is equally complex with propositional calculus of Bertrand Russel and Alfred North Whitehead in their famous book Principia Mathematica who has five axiom and one inference rule.

Fortunately, I can simplify Existential Graph System basing it on the CONSISTENCY p->p as axiom and use the ITERATION (p)’q <-> (pq)’q as the only inference rule generating other Peircean rules from the axiom. That discovery made me so happy, that I was blind to the simplest discovery by George Boole arithmetizing logic. In fact, Boole himself was blind to the enormous power of his original mathematical symbolization that was forgotten by all logician and scientists today. I will call the arithmetic symbol system of logic as arithmologic.

Thanks God, by reusing the original Boolean intuitive math symbolism, I finally discover that all Boolean tautologies can be derived from the simplest single axiom, which is nothing but TRUE or 1, and the familiar arithmetical rules and arithmetical definition of logical operations. I was so happy with this discovery of arithmologic, so I was emotionally excited to relate it to the theology of monotheistic religion :).

Well, now in this blog I will show you that we can derive all Boolean tautologies from the simplest axiom: 0. Now, non-theists will be glad finding all logical truths is generated from 0 or VOID. :)

Boolean Arithmologic

In my blog article, I have shown that the generator of all tautologies in logic is the single simple axiom TRUE represented in Boolean algebra as 1.
NOT a is represented by 1-a.
a OR b is is represented by a+b.
a AND b = NOT (NOT a OR NOT b) is represented by 1-(1-a+1-b)=a-1+b
IF a THEN b = NOT a OR b is represented by 1-a+b

For example to prove the tautology a AND a = a do the following
1                                                          {the TRUE axiom}
=1-a+a                                                 {because a-a=0}
=1-a+1-1+a                                         {because 1-1=0}
=(1-a+1-a+a)-1+(1-a+a-1+a)              {because a-a=0}
=(1-(a-1+a)+a)-1+(1-a+(a-1+a))         {because a-a=0}
=(a AND a -> a) AND (a -> a AND a)  {definition of AND and IF}
=(a AND a = a)                                      {definition of = }

Peircean arithmologic

However, if we translate the existential graph system of Charles Peirce into mathematical symbolism, then VOID or TRUE is represented by 0, and the AND or VOID connective is represented by *. So, NOT a is represented by 1/a because 1 is FALSE, and by the law of contradiction NOT a AND a = FALSE or a’*a = 1. Therefore, NOT a or a’ can be represented by 1/a.

It made me wonder, can we derived all tautologies from 0? Knowing that all arithmetical formulation of a tautology is equal to TRUE or 0. Reversing the arithmetical evaluation of a tautology can be regarded as a proof of it from the axiom 0 using arithmetic rules. Yes we can do it, provided that we know the strange logical arithmetic of Peircean logic.

In the the Peircean logical arithmetic 1/0 = NOT TRUE = FALSE = 1 and 1/1 = NOT FALSE = TRUE = 0
The logical equivalence a=b can be defined as bi-implication (a->b)*(b->a) and the implication a->b is defines as 1/(a*1/b)=NOT(a AND NOT a) – NOT a OR a. All tautologies f(a,b,c…) is TRUE is represented in Peircean logic as f(a,b,c,..)=0. So if we know the the Peircean logical arithmetic, we can derive f(a,b,c..) from 0.

For example x*x=x can be proven in the following manner
0                                                  {the TRUE axiom}
=1/1                                             {Peircean logical arithmetic}
=1/(x * 1/x)                                 {because x * 1/x = 1}
=1/x * 1/(1/x)                               {because 1/xy = 1/x * 1/y}
=1/(x*x*(1/x)) * 1/(x*(1/x*x))    {because x=(x*x)/x}
=(x*x->x) * (x->x*x)                    {definition of IF}
=(x*x=x)                                        {definition of = }

The Sommersian Arithmologic

Fortunately, I have lately found out the existence of another logical arithmetic which is simpler than the Peircean: the logical arithmetic of Fred Sommers. The Sommersian arithmetic has its own strangeness.
In the sommersian arithmologic 0 is representing both FALSE and TRUE and + representing both AND and OR. So NOT a is represented by -a, a AND b is represented by a+b and IF a THEN b or a->b is represented by -a+b.

So x+x=x can be proven in the following manner

0                                 {the TRUE axiom}
=-x+x                          {because -x+x=0}
=-(x+x)+x-x+(x+x)     {because -x+x=0}
=(x+x->x)+(x->x+x)   {definition of IF }
=(x+x=x)                     {definition of =  }

Afternotes:

(1) I think Sommersian arithmologic is the simplest axiom system in the foundation of logical algebra. It is simpler than Boolean and Peircean arithmologic which in turn simpler than Brownian, Kauffmanian dan Peircean pictorial algebra.

(2) In fact, the sommersian arithmologic can also be pictorially symbolized to get the simplest pictorial symbolization of logical algebra.

(3) Like the more complex pictorial system of Brown, Kauffman and Peirce, the sommersian arithmologic can be transformed into a very simple logic game of things, cards for example, that can be taught to pre-schoolers.

(4) Hopefully, this will revolutionize education by teaching arithmological algorithm in the logic game of things to solve many logical problems without sophisticated Boolean algebra.