## Monday, January 13, 2014

### On Logistic, 6 of 8: Logistic Fixedpoints

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:
x1 =   F1(x1, x2, … xn,…)
x2 =   F2(x1, x2, … xn,…)
xn =   Fn(x1, x2, … xn,…)
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 x4-2x2-1=0:
x = sqrt( 1 + sqrt(2) )

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 x3-x2-x-1=0:
x = ( cuberoot(19+sqrt(297)) + cuberoot(19-sqrt(297)) + 1 ) / 3

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 x4+x2-1=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) )