## Wednesday, May 16, 2018

### Diamond Bracket Forms and How to Count to Two; 3 of 10

From "Cybernetics & Human Knowing", Vol. 24 (2017), No. 3-4, pp. 161-188
2. Re-Entrance and Fixedpoints

Consider the Liar Paradox as a Brownian form:
______
|
L      =       L  |
|
This form contains itself. That can be represented via re-entrance, thus:
___________
|
L        =       |     |
|_____|

Let re-entrance permit any mark within a Brownian form to extend a tendril to a distant space, where its endpoint shall be deemed enclosed. This form sends a tendril into itself. Other re-entrant expressions include:
___________
_______  |
|  |
|  |  |
|_____|
_____________
_____     |
|  |  |
|  |  |  |
|_____|__|

_____________________
____________  |
|   ________   |  |
|          |   |  |
|      |   |   |  |
|      |_______|  |
|_________________|

__________________________
__________  _________  |
|          |  |
|  |  |    |  |  |  |
|  |  |____|  |  |  |
|  |__________|__|  |
|_____________|_____|

Self-reference can be expressed as a re-entrant brownian form, as a switching circuit, as a vector of forms, as an indexed list, and as a diamond-logic fixedpoint. For example:
___________
_______ |
| |                          Brownian form
|  | |
|____|

@
__________________________
/                          \        Switching Circuit
\________|\________|\______/     (triangles = “not” gates)
|/        |/          (wires and gates doubled)

@
___             ___
|           ____   |
|   A   =    B |   |
|           ____   |                          Brownian Form Vector
|   B   =    A |   |
|__              __|

@

A  =  [ [ A ]B ]A                                          Indexed List

@

(A , B) = (not B , not A)             Diamond-logic Fixedpoint