Monday, May 21, 2018

Diamond Bracket Forms and How to Count to Two; 6 of 10


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

Consider the liar paradox:
                           _______
                                 |
      A    =  not A    =      A  |     =    [ A ]A

                     _________
                                 |
Here is its Brownian form:       |___|

                          __________
Here it is as a circuit:       |        |
                          |___|\___|
                              |/

Here is its fixedpoint lattice:    i  -----------  j



Now consider Tweedle’s Quarrel:

Tweedledee: “Tweedledum is a liar.”
Tweedledum: “Tweedledee is a liar.”
                 _____   _________
      E    =       U |     _____ |  E = [[E]U]E
                 _____         | |
      U    =       E |       | | |
                             |___| 

                   __________________
Its circuit is:      |                 |
                   |____|\_____|\____|
                        |/     |/

    tf
   /  \
  /    \
This “toggle’s” lattice is:    ii       jj
  \    /
   \  /
    ft


Consider the following statement:
“This statement is both true and false.”
It resolves to this system, the “duck”:    B  =  [ [ B ]A B ]B

               _____        ___________b
       A   =     B |          ______a  |
               _____               |   |
       B   =    AB |            |  | | |
                                |____|_|


I call it the “Duck” because of the appearance of its circuit:
                 ____
                /    \     
        ___|\___\_|\_/
       /   |/     |/ \    
       \_____________/    


This is equivalent to the fixedpoint:
         B   =   ( B & ~ B )   =   dB  ;      a differential of itself!

Here is its lattice:     ii ----- tf ----- jj

The “triplet” has this form:      C  =  [ [ B C ]A [ C A ]B ]C

                ______      ______________________
      A     =    B C |       _________ _________ |
                ______               |         | |
      B     =    C A |         |  |  |   |  |  | |
                ______         |  |  |___|  |  | |
      C     =    A B |         |  |_________|__| |
                               |____________|____|

Three calling each other liars!

The triplet has this circuit:
                 _________ 
         _______/___      \
        /      /    \      \
        \__|\__\_|\_/__|\__/
        /  |/ \  |/  / |/  \
       /       \____/      /
       \__________________/

Its lattice is: 
                  tff
                /     \
               /       \
           iii -- ftf –- jjj
               \       /
                \     /
                  fft

Note that this lattice (called “M3”) is non-distributive:

a
                   <                 <             (a max b) min c  = 1 min c  =  c       
          0        <       b        <       1
                   <                 <             (a min c) max ( b min c) = 0 max 0 = 0     
c
On the other hand, it is “modular”:
x < z     implies    x max (y min z)  =  (x max y) min z
A theorem of lattice theory states that any non-distributive modular lattice contains M3 as a sublattice.


The “ant”, or “toggled buzzer”, has the form   C  =  [ [ [ B ]A ]B C ]C:
             ____        _______________
      A   =   B |             ________ |
             ____           | ______ | |
      B   =   A |           |      | | |
             ____           |   |  | | |
      C   =   BC|           |   |____| |
                            |__________|

The ant’s lattice is:           ftf
          <           <
iii                    jjj
     <             <
        tfi < tfj

Note that this lattice (called N5) is non-distributive:

                  b                           
         <               <          (a max b) min c  =  1 min c  =  c
0                               1  
       <                   <        (a min c) max ( b min c) = a max 0 = a
             a   <   c             

It is also non-modular:   a < c,  but  a max ( b min c ) =  a max 0  =  a;
         and  (a max b) min c    =  1 min c  =  c.

A theorem of lattice theory states that any non-distributive non-modular lattice contains N5 as a sublattice.
Now consider this Brownian form; “two ducks in a box”:

C       =       [   [a[a]]a   [b[b]]b  c  ]c

      _________________________________
           ___________   ___________  | a = [a[a]]
        |    _____   |     _____   |  |
        |        | | |         | | |  | b = [b[b]]
        |      | | | |       | | | |  |
        |      |___|_|       |___|_|  | c = [abc]
        |_____________________________|


In the nand interpretation, this is:    
a        =              Da                           =       “I am honest or a liar.”
b        =              Db                           =       “I am honest or a liar.”
c        =       ~ (a & b & c)                  =          “One of us is a liar.”

Note that sentence c is of the form
                   c         =         c -> ( da V db )
- which is Boolean only if the lower differentials disjoin to true.
It has this fixedpoint lattice:

     iti -------- ijt -------- tjj
    /   \                     /   \
   /     \                   /     \
iii       tti ----------- ttj       jjj
   \     /                   \     /
    \   /                     \   /
     tii -------- jit -------- jtj


In the nor interpretation, this is:
a        =                 da               =       “I am honest and a liar.”
b        =                 db               =       “I am honest and a liar.”
c        =       ~ ( a V b V c )        =       “All of us are liars.”
Note that sentence c is of the form:
                   c         =        ( Db & Db ) - c
- which is Boolean only if the upper differentials conjoin to false.
It has this fixedpoint lattice:

      ifi --------- ijf --------- fjj
     /   \                       /   \
    /     \                     /     \
   /       \                   /       \
iii         ffi ----------- ffj         jjj
   \       /                   \       /
    \     /                     \     /
     \   /                       \   /
      fii --------- jif --------- jfj

Note the fixedpoints ijf and jif; these are the only one where C has a boolean value; but this is due to Complementarity, an anti-boolean axiom. Without those points, this lattice would be modular and distributive; but with them it contains N5.