[CG] Re: Transformations of Logical Graphs

Transformations of Logical Graphs • 3 • https://inquiryintoinquiry.com/2024/05/07/transformations-of-logical-graphs… Re: Transformations of Logical Graphs • 1 • 2 • https://inquiryintoinquiry.com/2024/05/05/transformations-of-logical-graphs… • https://inquiryintoinquiry.com/2024/05/06/transformations-of-logical-graphs… We've been using the duality between entitative and existential interpretations of logical graphs to get a handle on the mathematical forms pervading logical laws. A few posts ago we took up the tools of groups and symmetries and transformations to study the duality and we looked to the space of 2‑variable boolean functions as a basic training grounds. On those grounds the translation between interpretations presents as a group G of order 2 acting on a set X of sixteen logical graphs denoting boolean functions. Last time we arrived at a Table showing how the group G partitions the set X into 10 orbits of logical graphs. Here again is that Table. Interpretive Duality as Group Symmetry • Orbit Order • https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-gr… I invited the reader to investigate the relationship between the group order |G| = 2, the number of orbits 10, and the total number of fixed points 16 + 4 = 20. In the present case the product of the group order (2) and the number of orbits (10) is equal to the sum of the fixed points (20). Is that just a fluke? If not, why so? And does it reflect a general rule? We can make a beginning toward answering those questions by inspecting the “incidence relation” of fixed points and orbits in the Table above. Each singleton orbit accumulates two hits, one from the group identity and one from the other group operation. But each doubleton orbit also accumulates two hits, since the group identity fixes both of its two points. Thus all the orbits are double‑counted by counting the incidence of fixed points and orbits. In sum, dividing the total number of fixed points by the order of the group brings us back to the exact number of orbits. Resources — Logic Syllabus • https://inquiryintoinquiry.com/logic-syllabus/ Logical Graphs • First Impressions • https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/ Logical Graphs • Formal Development • https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development… Regards, Jon cc: https://www.academia.edu/community/5M7moJ