Re: Futures Of Logical Graphs • Themes and Variations

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 -node operator, corresponding to an expression of the form is as follows.

• The rule for evaluating a -lobe operator, corresponding to an expression of the form is as follows.

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science