Algazel Principle

Algazel Principle
Armahedi Mahzar (c) 2015

Algazel

This is my surprising turn in my quest for logical simplicity. When I study Boolean logic I try to apply it to traditional logic of Aristotle. I was so disapointed when I read the validity proof of syllogism in George Boole book Laws of Thought, is very complex using strange terms like 1/0 or 0/0.

Aristotle saw his valid categorical syllogisms as implications. The Stoic saw hypothetical syllogisms and disjunctive syllogisms as implications. Implication is one way relation. It is not two way equivalence. Even Boole who wrote syllogistic premises and as equivalences did not know it.

Algazel Principle

However, the great Persian sufi al-Ghazali  in the 11th century, see all syllogisms as a ballance. In other words, the conjuction of premises and the conclusion are logical equivalent to each other. He logically interpreted the word al-mizan in the holy book of Islam, al-Quran, as syllogism.

When I first read about it, I could not believe it, because I thought that inference is one way directed, But, later on, after discovering Boolean arithmologic, I found out that al-Ghazali is indeed true. Valid syllogism, in fact, is an equivalence. Nothing wrong in it.

This principle of implication is so simple such that its discovery made me astonished, because it is so simple but also so powerful. I will call it as the Algazel Principle. In this blog I will show you by illustrate its aplication to classical logic.

Arithmologic coding

Arithmologic used arithmetical coding used by Boole in his algebra of logic. TRUE and FALSE is coded as 1 and 0 respectively. NOT x is coded as 1-x. x OR y is coded as addition x+y.

Boole coded x AND y as multiplication xy. But, using De Morgan Law, we can defined x AND y as NOT(NOT x OR NOT y) which can be coded as  1-(1-x+1-y)=x-1+y. So in arithmologic, AND is coded as -1+.

The interesting point with this arithmetic coding, the three fundamental principles of logic are becoming just algebraic identities. The principle of identity becomes x=x, the principle of excluded middle becomes x+1-x=1 and the principle of contradiction becomes x-1+1-x=0.

Proving Algazel Principle

With this arithmologic the valid syllogism IF (all m is p) AND (all s is m) THEN (all s is p) is coded as 1-((1-m+p)-1+(1-s-m)+(1-s+p) which can immediately evaluated as 1 or TRUE. If it is not evaluated to 1 then it is not valid.  

In general, a valid syllogism is a tautology. This fact can be used to prove Algazel principle as follow. If IF a AND b THEN c = TRUE is a valid syllogism, then its arithmologic coding is 1-(a-1+b)+c=1 which is equal to 1+c=1+(a-1+b) or c=a-1+b which is the code of c IS (a AND b).

So, logically, the syllogism is an equivalence. The conclusion IS the conjunction of its premises. It is so general, so it can be used to prove all stoic syllogisms is also equivalences as it is claimed by Algazel: the hypothetical and disjunctive.

Afternote

In fact, Algazel principle is not only true for just syllogisms, it is true for any tautological inference for example the Leibniz Praeclarum Theorema and the Peirce Law. Proof: the tautological  implication IF p THEN q is TRUE is coded as 1-p+q=1 or p=q which is an equivalence of consequent and antecedent.

The mystery is that Algazel did not know arithmologic, but in fact he know syllogism is equivalence. How could he know it? I think because he is a mystic and he know it by his mystical intuition seeing directly the Plato's world of mathematical ideas. So, anyway it seems to me that mysticism and mathematic are not contradicting each other.

The surprising fact is that so many mathematicians and logicians such as Aristotle, the Stoic logician, Leibnitz, Boole and Peirce did not recognize the Algazel Principle. It make me think that mystical contemplation can help mathematician or logician to discover more truths.

Ikhwan as-Safa 

 

Ikhwan as-Safa is a Muslim sect, popularly known as the “Brethren of Purity” who published the encyclopedia Rasa’ il (ca. 989), a forty-eight volume series.

This sect, which was centered in Basra—a thriving seaport at this time and a marketplace of foreign influences—believed in the purifying power of knowledge. Ikhwanian writings and teachings represented a Gnostic effort to reconcile Hellenistic beliefs with the teachings of the Quran. Pythagorean and Neoplatonic mystical concepts were dominant in the Brethren’s rationales.

They viewed geometric shapes as personalities bearing special attributes. The triangle represented harmony. The square represented stability and promoted the Four Element theory and thus a special status for the number 4 (reminiscent of the Chinese cosmological reverence for the number five). Of this fourfold division of Nature, the Ikhwans wrote:

God himself has made it such that the majority of the things of Nature are grouped in four such as the four physical natures which are hot, cold, dry and moist; the four elements which are fire, air, water and earth; the four humours which are blood, phlegm, yellow bile and black bile; the four seasons . . . , the four cardinal directions . . . , the four winds . . . , the four directions envisaged by their relation to the constellations; the four products which are the metals, plants, animals and men.( Nasr, An Introduction to Cosmological Doctrines, p. 50. For a more detailed discussion of Ikhwanian beliefs see Seyyed Hossein Nasr, Science and Civilization in Islam (Cambridge, MA: Harvard University Press, 1968), 152–57)

For the Ikhwans, as for the Pythagoreans, number appeared to be the key for understanding the relationships upon which the physical and spiritual world functioned. They developed their own system of abjad for the numerical symbolism of letters and devised numerical categories to clarify relationships.

In particular, the Brethren divided all beings and objects into nine states, since 9, the last digit in the decimal numeration system, closes a cycle and symbolically brings an end to the series of numbers. The categories are as follows:

The process of creation is divided twofold: first, God creates ex nihilo the Intellect; immediately after the Intellect’s emanation (fayd), it proceeds gradually, giving shape to the present universe. The order and character of emanation are described below. (rasa’il Ikhwan al-Safa’, vol. 1 p. 54; cf. rasa’il Ikhwan al-Safa’, vol. 3 pp. 184, 196-7; 235)

(1) Al-Bari’ (Creator, or God) is the First and only Eternal Being, no anthropomorphic attribute is to be ascribed to Him. Only the will to originate pertains to Him. The Ikhwan present an Unknowable God (Deus Absconditus) at the top of the hierarchy while the Qur’anic God (Deus Revelatus), another facet of God, guides people on the right path.

(2) Al-’Aql (Intellect or Gr. Noûs) is the first being to originate from God. It is one in number as God Himself is One. God created all the forms of subsequent beings in the Intellect, from which emanated the Universal soul and the first matter. It is clear, in the opinion of the Ikhwan, that the Intellect, a counterpart of God, is the best representative of God.

(3) Al-Nafs al-Kulliyya (The Universal Soul) is the Soul of the whole universe, a simple essence which emanates from the Intellect. It receives its energy from the Intellect. It manifests itself in the sun through which is animated the whole sublunary (material) world. What we call creation, in our physical world, pertains to the Universal Soul.

(4) Al-Hayula al-Ula (Prime Matter, arabicized from Gr. hyle), is a spiritual substance that is unable to emanate by itself. It is caused by the Intellect to proceed from the Universal Soul which helps it to emanate and accept different forms.

(5) Al-Tabi’at (Nature) is the energy diffused throughout all organic and inorganic bodies. It is the cause of motion, life, and change. The influence of intellect ceases at this stage of Nature. All subsequent emanations tend to be more and more material and defective.

(6) Al-Jism al-Mutlaq (The Absolute Body) comes about when First matter acquires physical properties, and it is the physical substance of which our world is made.

(7) The World of the Spheres (of the fixed stars, Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon) appears in the seventh stage of emanation. All the heavenly bodies are made up of a fifth element (ether), and are not subject to generation and corruption.

(8) The Four Elements (fire, air, water, and earth) come immediately under the sphere of the moon where they are subjected to generation and corruption. The Ikhwan adopted the view of Thales (d. c. B.C.E. 545) and the Ionians that the four “elements” change into one another, water becomes air and fire; fire becomes air, water, earth, etc.

(9) The Three Kingdoms are the last stage of emanation. The three kingdoms (mineral, plant, and animal) are made of proportional intermixture of the four elements.

 

The first four numbers in this series and the entities in the first four categories were, by the Pythagorean concept of tetractys, considered to be pure, universal beings since, within their sum, they contained all numbers. The other beings are compound objects.

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.

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.