Iterated nand - Printable Version +- Tetration Forum ( https://math.eretrandre.org/tetrationforum)+-- Forum: Tetration and Related Topics ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)+--- Forum: Mathematical and General Discussion ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)+--- Thread: Iterated nand ( /showthread.php?tid=1159) |

Iterated nand - Xorter - 03/12/2017
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.) RE: Iterated nand - tommy1729 - 03/23/2017
(03/12/2017, 03:08 PM)Xorter Wrote: I am interested in iterates (and uniterates) of logical operators, like nand, xor ... etc. . RE: Iterated nand - Xorter - 03/27/2017
(03/23/2017, 01:27 PM)tommy1729 Wrote: --- Very good questions. Well, to be honest, (x or not x), (x and not x) are neither trivial nor paradoxical, just think of fuzzy logic with x=0.5 (half truth or sg like this), then each expression results 0.5, right? What is the half iterative of negation? The imaginary negation, about what we know that: ¬^.5 o ¬^.5 = ¬, where o is composition. It is as mysterious as the imaginary unit. I think. But it solves your so-called paradox, that: "This statement is false.". Its value neither true nor false, it is (¬^.5)(true), so so-called this is the imaginary truth ... solving the all the paradoxes. What is the pi-th and the i-th iterate of the logical ops? This is my question. |