## 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.