From "Cybernetics & Human Knowing", Vol. 24 (2017), No. 3-4, pp. 161-188
4. Fixedpoint Lattices
Any diamond-logic function F(x) has the extreme fixedpoints: Fn(i), Fn(j), the leftmost and rightmost fixedpoints. But often this is not all.
In general, F has an entire lattice of fixedpoints.
Lemma: For any diamond-logic f;
f(x) min f(y) > f(x min y)
f(x) max f(y) < f(x max y)
Proof is an exercise for the student.
Theorem: If F is diamond-logic, and has fixedpoints a and b, then these fixedpoints exist:
a minF b = the rightmost fixedpoint left of both a and b = F2n(a min b)
a maxF b = the leftmost fixedpoint right of both a and b = F2n(a max b)
Proof. Let a and b be fixedpoints, and let c be a fixedpoint left of both a and b. Then (a min b) > c ; so (a min b) = F(a) min F(b) > F(a min b) > F(c) = c
Ergo (a min b) is a left seed greater than c:
(a min b) > F(a min b) > F2(a min b) > ...
> F2n(a min b) = F(F2n(a min b)) > c
Therefore F2n(a min b) is a fixedpoint left of a and of b, and is also the rightmost such fixedpoint. Thus, F2n(a min b) = a minF b . QED.
Similarly, F2n(a max b) = a maxF b . QED.
For instance, consider the following Brownian form:
| _____a _____b | |
| | | | |
| | |____| | | |
| |________| | |
This is equivalent to this bracket-form system:
a = [b] ; b = [a] ; c = [ab] ; d = [cd] .
In the standard interpretation, (a,b,c,d) is a fixedpoint for:
F(a,b,c,d) = ( ~ b, ~ a, ~(a Ú b), ~(c Ú d) )
In the nand-gate interpretation:
d = ~(d Ù c) = ~d Ú ~c = (d Þ da )
Sentence d says “If I’m not mistaken, then A is both true and false”.
In the nor-gate interpretation:
d = Da - d ;
Sentence d says “A is true or false, and I am a liar.”
F has this fixedpoint lattice:
tffi ----- tffj
ftfi ----- ftfj
To find minF of tffj and ftfj, first take their minimum, then apply F three times: (tffj min ftfj) = iifj è iiij è iiit è iiii