[CG] Transformations of Logical Graphs

Transformations of Logical Graphs • 1 • https://inquiryintoinquiry.com/2024/05/05/transformations-of-logical-graphs… Re: Interpretive Duality in Logical Graphs • 1 • https://inquiryintoinquiry.com/2024/04/22/interpretive-duality-in-logical-g… Re: Mathematical Duality in Logical Graphs • 1 • https://inquiryintoinquiry.com/2024/05/03/mathematical-duality-in-logical-g… All, Anything called a “duality” is naturally associated with a transformation group of order 2, say a group G acting on a set X. Transformation groupies generally refer to X as a set of “points” even when the elements have additional structure of their own, as they often do. A group of order two has the form G = {1, t}, where 1 is the identity element and the remaining element t satisfies the equation t² = 1, being on that account self‑inverse. A first look at the dual interpretation of logical graphs from a group-theoretic point of view is provided by the Table below. Interpretive Duality as Group Symmetry • https://inquiryintoinquiry.files.wordpress.com/2021/02/peirce-duality-as-gr… The sixteen boolean functions f : B × B → B on two variables are listed in Column 1. Column 2 lists the elements of the set X, specifically, the sixteen logical graphs γ giving canonical expression to the boolean functions in Column 1. Column 2 shows the graphs in existential order but the order is arbitrary since only the transformations of the set X into itself are material in this setting. Column 3 shows the result 1γ of the group element 1 acting on each graph γ in X, which is of course the same graph γ back again. Column 4 shows the result tγ of the group element t acting on each graph γ in X, which is the entitative graph dual to the existential graph in Column 2. The last Row of the Table displays a statistic of considerable interest to transformation group theorists. It is the total incidence of “fixed points”, in other words, the number of points in X left invariant or unchanged by the various group actions. I'll explain the significance of the fixed point parameter next time. Regards, Jon cc: https://www.academia.edu/community/l7jBGO