Diamond Bracket Forms and How to Count to Two; 2 of 10

From "Cybernetics & Human Knowing", Vol. 24 (2017), No. 3-4, pp. 161-188

1.  Diamond Bracket Forms

This paper uses brackets [] to denote G Spencer-Brown’s crossing symbol. The arithmetic initials are then:

[ ] [ ]  =   [ ] .

[ [ ] ]  =       .

If we call []  “1” and  [[]]  “0”, then we get these equations:

[ 0 ]  =  1    ;    [ 1 ]  =  0    ;

0 0   =   0    ;    0 1   =  1 0   =   1 1   =   1  .

The algebraic initials are:

[ [ a ] [ b ] ] c    =   [ [ a c ] [ b c ] ]     .

[ [ a ] a ]           =                                 .

We can identify bracket forms with boolean logic this way:  

[ ]                                            =       true ;

                   [ [ ] ]                                        =       false ;  

                   [ A ]                                         =       ~ A ;       

                   A  B                                        =       A Ú B  ;

                   [ AB ]                                                =       A nor B

          That is the “nor-gate” interpretation. Here’s one for “nand-gate”:

[ ]                                            =       false ;

                   [ [ ] ]                                        =       true ;  

                   [ A ]                                         =       ~ A ;       

                   A  B                                        =       A Ù B  ;

                   [ AB ]                                                =       A nand B

          This paper extends the bracket algebra to four-valued “diamond logic” by introducing two complementary paradox forms, 6 and 9, with these rules:

[ 6 ]  =  6  ;   [ 9 ]  =  9  ;   6 9  =  [ ]  .

          Assume that [] dominates juxtaposition:    x []    =   []

That implies these equations:

10 = 16 = 19 = 11 = 1;   [1] = 0 ;

91 = 96 = 1 ; 99 = 90 = 9 ;  [9] = 9 ;

61 = 69 = 1 ; 66 = 60 = 6 ;  [6] = 6 ;

00 = 0 ; 06 = 6 ; 09 = 9 ; 01 = 1 ;  [0] = 1 .

The bracket forms 6 and 9 can be interpreted as “underdetermined” and “overdetermined”; where “underdetermined” means “insufficient data for definite answer”, and “overdetermined” means “contradictory data”. An underdetermined statement is neither provable nor refutable, and an overdetermined statement is both provable and refutable.

Underdetermined can also be called “gap” - i.e. neither true nor false; and overdetermined can be called “glut” - i.e. both true and false. Therefore:

True is true   =   true;              True is false  =   false;

False is true   =   false;             False is false  =   true;

Gap is true    =   false;             Gap is false   =   false;

Glut is true    =   true;              Glut is false   =  true.

Assume that values are equal if they are equally true and equally false:

If       (A is true) = (B is true)      and     (A is false) = (B is false)

Then      A = B.

We can then define the logical operators thus:

(A Ù B) is true    =    (A is true) Ù (B is true)

(A Ù B) is false   =    (A is false) Ú (B is false)

(A Ú B) is true    =    (A is true) Ú (B is true)

(A Ú B) is false   =    (A is false) Ù (B is false)

(~ A) is true        =    (A is false)

(~ A) is false       =    (A is true)

These definitions imply this table:

x:  ~ x:      Ù y:             Ú y:     

                          t  f  gp gl      t  f  gp gl  

t     f       t  f  gp gl      t  t  t  t 

f     t       f  f  f  f       t  f  gp gl   

gp    gp      gp f  gp f       t  gp gp t   

gl    gl      gl f  f  gl      t  gl t  gl   

This is Belnap’s 4-valued relevance logic. It is equivalent to diamond bracket forms under four interpretations:

[] = T;  [[]] = F;  6 = Gap;  9 = Glut ;  [XY]  =  X nor Y

[] = T;  [[]] = F;  6 = Glut;  9 = Gap ;  [XY]  =  X nor Y

[] = F;  [[]] = T;  6 = Gap;  9 = Glut ;  [XY]  =  X nand Y

[] = F;  [[]] = T;  6 = Glut;  9 = Gap ;  [XY]  =  X nand Y

Both are equivalent to diamond logic, which has these tables:

x:  ~ x:    Ú y:    

            t f i j 

    

          t    f     t t t t   

f     t     t f i j   

i     i     t i i t

j     j     t j t j

Diamond logic operates on pairs of Boolean values a/b, where t = t/t,  f = f/f,  i = t/f,  j = f/t,  ‘Ú’ acts termwise, and ‘~’ acts with a twist:

(a/A) Ú (b/B)                  =       (a Ú b) / (A Ú B)

    ~ (a/A)                       =         (~ A) / (~ a)

Likewise, you can define diamond-logic circuitry on pairs of wires, with ‘or’ gates matching wire to wire, and ‘not’ gates operating with a twist.

         

The four diamond bracket forms 0, 1, 6, and 9 have these bracket identities:

          Transposition:             [ [ a ] [ b ] ] c         =    [ [ a c ] [ b c ] ] 

          Occultation:                   [ [ a ] b ] a            =              a

          Complementarity:     [ [ 6 ] 6 ] [ [ 9 ] 9 ]    =             []   

Call these the diamond bracket axioms. Also, assume commutativity and associativity for juxtaposition:

a  b   =   b a       ;       a b   c     =     a    b c

These equations are implicit in the bracket notation. Brackets distinguish only inside from outside, not left from right.

The diamond bracket axioms imply the following theorems:

Majority.   [[xy][yz][zx]]         =       [[x][y]] [[y][z]] [[z][x]]

Let M(x,y,z) denote both of these forms. Then:

Transmission.    [ M(x,y,z) ]    =    M([x],[y],[z])

Distribution.     x M(a,b,c)      =       M(xa, b, xc)

Redistribution.    [[x][M(a,b,c)]]       =       M([[x][a]], b, [[x][c]])

General Distribution.  M(x,y,M(a,b,c))    =   M(M(x,y,a) , b, M(x,y,c))

Coalition.    M(x, x, y)      =      x

Opposition.    M(x, t, f)      =      M(x, i, j)     =    x

General Associativity.  M(x,a,M(y,a,z))    =   M(M(x,a,y), a, z)

Proofs are an exercise for the student.

Majority defines four connectives; Ù, Ú, min and max. M(x,F,y) = x Ù y;  M(x,T,y) = x Ú y; define M(x,I,y) = x min y;  M(x,J,y) = x max y.

x:  Ú y:      Ù y:      min y:    max y:    

     t f i j   t f i j   t f i j   t f i j

    

          t    t t t t   t f i j   t i i t   t j t j

f    t f i j   f f f f   i f i f   j f f j

i    t i i t   i f i f   i i i i   t f i j  

j    t j t j   j f f j   t f i j   j j j j

Ù, Ú, min and max have these properties: associativity; recall; attractors ( F, T, I and J, respectively);  identities (T,F,J and I); mutual distribution; and also:

De Morgan: ~(xÙy) = ~xÚ~y  ;     ~(xÚy) = ~xÙ~y

Transmission: ~(x min y) = ~x min ~y  ;     ~(x max y) = ~x max ~y

          Min and max define “phase order”:

x < y        iff         x min y  =  x        iff         x max y  =  y

T

<          <

I          <          J

<          <

F

Phase order is a knowledge ordering in the glut-gap interpretation.

“And”, “or” and “not” distribute over min and max; therefore:

<  is preserved by disjunction, conjunction and negation:

a < b     implies    a Ú c   <   b Ú c

  and           a Ù c   <   b Ù c

  and                 ~ a  <  ~ b

<  is preserved by any diamond-logic function:

a < b     implies    F(a)  <  F(b)

This follows by induction from the previous result.

The dual paradoxes i and j define “but”, the “junction” operator:

x / y     =     (x Ù i) Ú  (y Ù j)     =     (x Ú  j) Ù (y Ú  i)

x:  / y:    

     t f i j   

          t    t i i t   

f    j f f j   

i    t i i t   

j    j f f j   

Here are the junction laws:

          Recall:                   a/a                       =                 a

          Polarity:           (a/A)/(b/B)               =                 a/B

          Parallellism:   (a/A) Ù  (b/B)             =         (a Ù b) / (A Ù B)

                                 (a/A) Ú  (b/B)              =         (a Ú  b) / (A Ú B)

                             M( a/A, b/B, c/C )                    =       M(a,b,c) / M(A,B,C) 

          Reflection:          ~ (a/A)                   =              (~ A)/(~ a)

          Brackets:          [(a/A)(b/B)]              =               [AB]/[ab]           

          Positives:              x   Ù  y                  =       (x max y) / (x min y)

                                      x   Ú  y                  =       (x min y) / (x max y)

                                      x  min  y               =       (x Ú y) / (x Ù y)

                                      x  max  y              =       (x Ù y) / (x Ú y)

          Positives, Parallelism and Recall imply Lattice Majority:

M(x,y,z)  =  (x min y) max (y min z) max (z min x)

M(x,y,z)  =  (x max y) min (y max z) min (z max x)

          Lattice Majority implies Mediocrity:   If a < b < c   then M(a,b,c)  =  b

          Proof. M(a,b,c)     =   (a min b) max (b min c) max (c min a)

                                      =   a max b max a    =   b

          Bracketing: If a<b<c   then   bÚ(aÙc)   =   bÙ(aÚc)    =    b

          Proof. bÚ(aÙc)  =  bÚ(bÙa)Ú(bÙc)Ú(aÙc)  =  bÚM(a,b,c)  = bÚb  = b

                   bÙ(aÚc)  =  bÙ(bÚa)Ù(bÚc)Ù(aÚc)  =  bÙM(a,b,c)  = bÙb  = b

          Differential Normal Forms:

Any diamond-logic function F(x) can be put into these forms:

F(x)   =       (F(t) Ù x)  Ú  (F(f) Ù ~ x)  Ú   M(F(i), dx, F(j))

F(x)   =       (F(t) Ú ~ x)  Ù  (F(f) Ú x)  Ù  M(F(i), Dx, F(j))

where  dx  = (xÙ~x)  and Dx = (xÚ~x) 

This defines the function in terms of its values.

Proof is by cases, plus Bracketing, Mediocrity and these phase relations:

i < F(i) < F(t) < F(j) < j   and   i < F(i) < F(f) < F(j) < j  .

          You can also prove this theorem from the diamond bracket axioms. For details, see my book, “Diamond, a Paradox Logic”.

The differential normal forms imply Completeness:

Any equational identity in diamond can be deduced from the bracket axioms.

Proof is by induction on the number of variables.