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.
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