**6. Logistic Fixedpoints**

Let F
denote the generic finite positive quantity. The triple 0,F,∞ are then logistic ‘seen at low-resolution’, for all finite positives are confused. In
this low-rez view, here are tables for addition, reduction, Einstein
addition and reciprocal:

**+ 0 F ∞ {+} 0 F ∞ ~ 0 F ∞ 1/x**

**0 0 F ∞ 0 0 0 0 0 0 F ∞ ∞**

**F F F ∞ F 0 F F F F F F F**

**∞ ∞ ∞ ∞ ∞ 0 F ∞ ∞ ∞ F 0 0**

These are isomorphic to
‘and’, ‘or’, ‘iff’ and ‘not’, in three-valued Kleenean logic, if we identify 0
with True, Finite with Intermediate, and ∞
with False. So logistic, at low resolution, looks Kleenean.

Kleenean
logic has a fixedpoint property:

The
following self-referential system, if Kleenean, has a solution:

x

_{1}= F_{1}(x_{1}, x_{2}, … x_{n},…)
x

_{2}= F_{2}(x_{1}, x_{2}, … x_{n},…)
…

x

_{n}= F_{n}(x_{1}, x_{2}, … x_{n},…)
…

We
can find such a fixedpoint by iteration. Start by setting all the x’s to the
intermediate truth value I; evaluate the functions; substitute these values
back into the x’s; repeat. The iteration will converge to a fixedpoint in
finitely many steps if the system is finite. If the system is infinite, then it
converges after sufficiently large-ordinal transfinite iteration.

Self-referential
Kleenean systems have solutions; logistic is Kleenean at low resolution;
therefore logistic self-refers at low-rez. Does it self-refer at high-rez?

Define
a logistic function as “positive” if it is defined from variables, constants,
addition and reduction only; no reciprocals. Then you can prove by induction
that positive logistic functions preserve linear order; and this implies that
any self-referential system of positive logistic functions has a fixedpoint.
Just iterate from (0,0,0,…); each component must increase with every step; this
converges to a fixedpoint in the closed interval [0, ∞]. Therefore

**positive logistic systems of any dimension can self-refer.**This implies, among other things, infinite sums and reductions.
Full logistic
systems, which include reciprocal as well as the additions, do not preserve
linear order; nonetheless they can still self-refer. The interval [0,∞] is compact and convex; so are finite
powers of it; so by Brouwer’s theorem,

**any finite-dimensional logistic system can self-refer.**If the system is infinite-dimensional, then a fixedpoint exists by Tychonoff’s theorem.
Here
are some logistic systems:

x = 1/x

“This
sentence is false.”

It
has a solution: x = 1.

x = x+1

“This
sentence and the Liar are true.”

Solution:
x = ∞. Infinite, or false.

x = x{+}1

“This
sentence or the Liar are true.”

Solution:
x = 0. Void, or true.

x = 1 + 1/x

“The
Liar is true and this sentence is false.”

Solution:
x = the golden mean, about 1.6108;
therefore falser than the Liar.

x = 1 {+}
1/x

“The
Liar is true or this sentence is false.”

“This
sentence implies the Liar.”

Solution:
x = 1/phi, about 0.6108; therefore truer
than the Liar.

x = 4(x
{+} 1/x)

= (x{+}1/x)+ (x{+}1/x)+ (x{+}1/x)+ (x{+}1/x)

“This
sentence is true or false, and this sentence is true or false, and this
sentence is true or false, and this sentence is true or false.”

Solutions:
x = zero, or root 3 (about 1.732).

x = (x
+ 1/x)/4

= (x+1/x){+}(x+1/x){+} (x+1/x){+} (x+1/x)

“This sentence
is true and false, or this sentence is true and false, or this sentence is true
and false, or this sentence is true and false.”

Solutions:
x = infinity, or root 1/3 (about 0.577).

x = (x
+ 2/x) / 2

= (x +
1/x + 1/x) {+} (x + 1/x + 1/x)

“This
sentence is true, and it is false, and it is false,

or

this
sentence is true, and it is false, and it is false.”

Solutions:
x = infinity, or root 2 (about 1.414).

x = 2(x
{+} 1/(2x))

= (x {+}
1/x {+} 1/x) + (x {+} 1/x {+} 1/x)

“This
sentence is true, or it is false, or it is false,

and

this
sentence is true, or it is false, or it is false.”

Solutions:
x = zero, or root ½ (about 0.7071).

x = 1/x
+ (x~x)

= 1/x +
(x{+}1/x) + (x{+}1/x)

“This
sentence is false, and it is equivalent to itself.”

“This
sentence is false, and it is true or false, and it is true or false.”

Solution:
same as the solution to x

^{4}-2x^{2}-1=0:
x
= sqrt( 1 + sqrt(2) )

About 1.553773974

x = 1
+ (x~x)

= 1 +
(x{+}1/x) + (x{+}1/x)

“The
Liar is true, and this sentence is equivalent to itself.”

“The
Liar is true, and this sentence is true or false, and it is true or false.”

Solution:
same as the solution to x

^{3}-x^{2}-x-1=0:
x
= ( cuberoot(19+sqrt(297)) + cuberoot(19-sqrt(297)) + 1 ) / 3

About 1.839286755

x = 1/x {+} (x + 1/x)

“This sentence
is false, or it is true and false.”

“If this
sentence is true, then it is true and false.”

Solution: same
as the solution to x

^{4}+x^{2}-1=0:
x
= sqrt(1/phi), where phi is the golden
mean

x is about 0.7861513778

Here is a
two-component fixedpoint:

A
= 1/A {+} 1/B

B = 1/B {+} A

Tweedledee: “If I’m not mistaken, then Tweedledum is mistaken.”

Tweedledum: “If I’m not mistaken, then Tweedledee is not mistaken.”

Iteration solution: (0.7653668647... , 0.5411961001...)

This matches the algebraic solution:

A = sqrt( 2 - sqrt(2) )

B = sqrt( 1 - 1 / sqrt(2) )

B = 1/B {+} A

Tweedledee: “If I’m not mistaken, then Tweedledum is mistaken.”

Tweedledum: “If I’m not mistaken, then Tweedledee is not mistaken.”

Iteration solution: (0.7653668647... , 0.5411961001...)

This matches the algebraic solution:

A = sqrt( 2 - sqrt(2) )

B = sqrt( 1 - 1 / sqrt(2) )

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