Simplifying Primary Algebra

Simplifying Primary Algebra

George Spencer-Brown

   is a master of axiomatic logical algebra. He managed to reduce the five Principia axioms for propositional algebra to just two axioms: 

or,  wriiten in parentheses notation,

position (x(x))=  and
transposition ((xz)(yz))=((x)(y))z,

Reading it existentially, the first axiom is the logical identity x->x and the second axiom is the distribution OR over AND.n of the principle 

Louis Kauffman 

  simplified the Brownian algebra by taking the sixth consequence, extension

or  ((x)y)((x)(y))=x,  as the single axiom for the algebra. It is a beautiful axiom, because reading it existentially, it is the algebraic formulation  the ancient principle of Reductio ad Absurdum: a proposition x is true if only if its negation, NOT x, implies y and NOT y, a contradiction.

However, by defining x=y as x IFF y = (IF x THEN y) AND (IF y THEN X) or (x(y))((x)y), we can derive transposition axiom as a theorem from a new simpler single axiom: position.

Proof:
Transposition ((xz)(yz))=((x)(y))z
=( (((xz)(yz)))((x)(y))z )( ((xz)(yz))(((x)(y))z) )     definition of =
=( (((x )(y )))((x)(y)) )( ((x )(y ))(((x)(y)) ) )           replacement z=
=( (  u         )       u  )(       u     (    u      ) )           substitution u=((x)(y))
=                                                                         position

I think position is simpler than extension because it reduce the number of variables from 2 to 1 and the crosses from 5 to 2.

The question is: can we make another simplification?