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