Iterated nand

[tommy1729]

I am interested in iterates (and uniterates) of logical operators, like nand, xor ... etc. .
Let's investigate the nand op:
x↑y = ¬(x&y)
Iterating x O y means this: x O x O ... O x (y-times)
Thus
x ↑ x ↑ ... ↑ x := x ↑↑ y
x ↑↑ 1 = x
x ↑↑ 2 = x ↑ x = ¬x
x ↑↑ 3 = x ↑ ¬x = ¬(x & ¬x) = x v ¬x
x ↑↑ 4 = x ↑ (x v ¬x) = ¬(x & (x v ¬x)) = x & ¬x
x ↑↑ 5 = x ↑ (x & ¬x) = ¬(x & (x & ¬x)) = x v ¬x
x ↑↑ 6 = x & ¬x
x ↑↑ 7 = x v ¬x
... etc.
So
x ↑↑ 2k = x & ¬x
x ↑↑ 2k-1 = x v ¬x
where k is bigger integer than 2
My question: Can k be any real or complex number?
(In my view to do this, we should know what "between" & and v is.)

---

I like the basic idea to do dynamics in set theory , Logic and the alike.

However i see many issues.

First you write alot " x and not x " and "x or not x" and they are ( in boolean ) either trivial or paradoxical !
Like " this sentense is false ".
or " this is true and false " , " this is true or false.

Secondly

suppose we let times -1 mean not.


Than iterations of not give the unit circle in the complex plane.

but what does it mean " i " ? What does it mean to half-iterate NOT ?

0r the pi th iteration of x OR y ???

you only have a few things like and or not true etc.
but you want continue iterations ??

or if you introduce new things like " i " above you need to Well define it !!

so as of now im very skeptical for continue iterations.

as for integer iterates that might work.

i think one then needs to associate it with groups or rings.

maybe modular arithmetic too.

Regards

tommy1729
the master