Proving Nicod Axiom

Proving Nicod Axiom
Armahedi Mahzar (c) 2018

Jean Nicod
(1893-1924)

In my last blog, I have proven that all axioms of the propositional calculus

✸1.2. ⊦: p ∨ p .⊃. p                               principle of tautology
✸1.3. ⊦: q .⊃. p ∨ q                               principle of addition
✸1.4. ⊦: p ∨ q .⊃. q ∨ p.                       principle of permutation
✸1.5. ⊦: p ∨ ( q ∨ r ) .⊃. q ∨ ( p ∨ r )  associative principle
✸1.6. ⊦:. q ⊃ r .⊃: p ∨ q .⊃. p ∨ r         principle of summation

in Principia Mathematica are the consequences of the primary algebra in the book Laws of form.

However, in his the second edition of Principia Mathematica, Bertrand Russell wrote that his 5 axioms, which used 2 logical operations, can be derived from a single axiom

DDpDqrDDtDttDDsqDDpsDps

discovered by Jean Nicod using single logical operation NAND, where x NAND y is written as Dxy in Polish notation and a single rule of inference. ( see https://projecteuclid.org/download/pdf_1/euclid.ndjfl/1093958259   )

Now, in this letter I will try to show you that Nicod's axiom and rule can also be derived from primary algebra, existentially read.

Nicod system:

Nicod single operation of incomptability x NAND y is symbolized by x|y. However we can rewrite x|y using Brown-Peirce parenthesis [xy]. Using this notation his system can be rewritten as

Axiom:  [ [p[qr]] [[t[tt]][ [sq] [[ps][ps]] ]]]   
Rule:   from p and [p[qr]] we can get q 

In the box notation  we get



We can prove Nicod axiom and rule can be proven as the consequences of the Brownian Primary algebra using its first axiom (position), first three consequences (reflection, generation and integration):and the fifth consequence (iteration).

Position    : [p[p]] =
Reflection :   [[p]] = p,
Generation : [pq]q = [p]q and
Integration:    [ ]p  = [ ]
Iteration          p p  =  p

Nicod Axiom

Nicod Axiom can be proven as a consequence of the primary algebra.: 
Proof:



Nicod rule

Nicod used a generalized modus ponens p[p[qr]]->q
as its inference rule. It can be rewritten as [p[p[qr]][q]]
Now it can be proven that is TRUE or equal to VOID
proof:
......


Afternotes
 
Nicod system, like Principia's true propositions are not equations. But by transforming them into an equations with the right hand side as TRUE or VOID, we can proved them as the consequences of Brownian algebraic primitives using the axiom and known consequences of the primary algebra

Nicod axiom is a very long complex formula. It can be replaced by a short formula of Huntington, called as extension by Spencer-Brown, which is proven by Louis Kauffman as the single axiom for the primary algebra.

If we add equivalence definition and algebraic identity as inference rule, then we can ultimately simplify the primary algebra using position as its single axiom and derive transposition as a consequence.

But, I might be wrong. If you can show mistakes in the proofs above,
Please show me

Proving Principia Axioms

Proving Principia Axioms

Principia Mathematica tried to construct the whole mathematics using calculus of proposition as its basic tool. The calculus itself is based on five simple axioms: tautology, addition, permutation, Association ad Summation. 




Based on this calculus, the whole Boolean algebra can be constructed. However, Boolean algebra can also be built on the base three axioms of Sheffer   which used only one logical NAND operation.

In the appendix of his book Laws of Form, George Spencer-Brown  has proven that the three Sheffer axioms can be proven as consequences of his two simple axioms: position and transposition.

This means that the five axioms of the Principia's calculus of proposition can be derived from the two axioms of brownian algebra that can be simulated by the game of things as it is shown in my last blog.

In this blog we will represent the Principia axioms into a configuration of boxes and colored balls and challenge you to transform them to a single box representing TRUE. If you can do it, then you already proved they are  the consequences of Brownian algebra.


  









My proof is using the following first three consequences of Brownian algebra considered as rules of transformations in the Brownian game of things



Tautology can be proved by using generation twice ended by doing integration.
Addition, permutation and association can be proved in a similar manner.
Summation can be proved firstly by removing the red ball within the box, then removing the contents of the first box using generation ended by doing integration.