[CG] Re: Operator Variables in Logical Graphs

Operator Variables in Logical Graphs • 11 • https://inquiryintoinquiry.com/2024/04/20/operator-variables-in-logical-gra… Re: Futures Of Logical Graphs • Themes and Variations • https://oeis.org/wiki/Futures_Of_Logical_Graphs • https://oeis.org/wiki/Futures_Of_Logical_Graphs#Themes_and_variations All, This post and the next wrap up the Themes and Variations section of my speculation on Futures of Logical Graphs. I made an effort to “show my work”, reviewing the steps I took to arrive at the present perspective on logical graphs, whistling past the least productive of the blind alleys, cul‑de‑sacs, detours, and forking paths I explored along the way. It can be useful to tell the story that way, partly because others may find things I missed down those roads, but it does call for a recap of the main ideas I would like readers to take away. Partly through my reflection on Peirce's use of operator variables I was led to what I called a “reflective extension of logical graphs”, amounting to a graphical formal language called the “cactus language” or “cactus syntax” after its principal graph‑theoretic data structure. The abstract syntax of cactus graphs can be interpreted for logical use in a couple of ways, both of which arise from generalizing the negation operator “( )” in a particular direction, treating “( )” as the controlled, moderated, or reflective negation operator of order 1 and adding another operator for each integer greater than 1. The resulting family of operators is symbolized by bracketed argument lists of the forms “( )”, “( , )”, “( , , )”, and so on, where the number of places is the “order” of the reflective negation operator in question. Two rules suffice for evaluating cactus graphs. • The rule for evaluating a k‑node operator, corresponding to an expression of the form “x₁ x₂ … xₖ₋₁ xₖ”, is as follows. Figure 16. Node Evaluation Rule • https://inquiryintoinquiry.com/wp-content/uploads/2019/07/box-xj-node-evalu… • The rule for evaluating a k‑lobe operator, corresponding to an expression of the form “(x₁ , x₂ , … , xₖ₋₁ , xₖ)”, is as follows. Figure 17. Lobe Evaluation Rule • https://inquiryintoinquiry.com/wp-content/uploads/2019/07/box-xj-lobe-evalu… Regards, Jon cc: https://www.academia.edu/community/V0EJZ7 cc: https://mathstodon.xyz/@Inquiry/112225263055943815