Logical Graphs • Discussion 8 • https://inquiryintoinquiry.com/2023/10/02/logical-graphs-discussion-8/ Re: Logical Graphs • Formal Development • https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development… Re: Laws of Form • Alex Shkotin • https://groups.io/g/lawsofform/message/2468 Hi Alex, I got my first brush with graph theory in a course on the Foundations of Mathematics Frank Harary taught at the University of Michigan in 1970. Frank was the don, founder, mover, and shaker of what we affectionately called the “MiGhTy” school of graph theory, spawned at U of M, Michigan State, Illinois, Indiana, and eventually spreading to other hotbeds of research in the Midwest and beyond. Later I took my first graduate course in graph theory from Ed Palmer at Michigan State, using Harary's “Graph Theory” as the text of choice. Definitions of graphs vary in style and substance according to the level of abstraction befitting a particular approach or application. The following is a classic formulation, one which covers the essential ideas in a very short space, and one whose elegance and power I've come to appreciate more and more as time goes by. Harary (1969): ❝A “graph” G consists of a finite nonempty set V = V(G) of p “points” together with a prescribed set X of q unordered pairs of distinct points of V. Each pair x = {u, v} of points in X is a “line” of G, and x is said to “join” u and v. We write x = uv and say that u and v are “adjacent points” (sometimes denoted u adj v); point u and line x are “incident” with each other, as are v and x. If two distinct lines x and y are incident with a common point, then they are “adjacent lines”. A graph with p points and q lines is called a (p, q) graph. The (1, 0) graph is “trivial”.❞ (Harary, Graph Theory, p. 9). I'll be hewing fairly close to that definition and terminology, though most graph theorists are used to the more common variations, like “nodes” instead of “points” and “edges” instead of “lines” — all but for the notion of “painted graphs” where I had to invent a new term on account of the fact that “labels” and “colors” were already taken for other uses. References — • Harary, F. (1969), Graph Theory, Addison-Wesley, Reading, MA. • Harary, F., and Palmer, E.M. (1973), Graphical Enumeration, Academic Press, New York, NY. • Palmer, E.M. (1985), Graphical Evolution : An Introduction to the Theory of Random Graphs, John Wiley and Sons, New York, NY. Regards, Jon

Logical Graphs • Discussion 10 • https://inquiryintoinquiry.com/2024/09/18/logical-graphs-discussion-10/ Re: Logical Graphs • Formal Development • https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development… Re: Laws of Form • Armahedi Mahzar • https://groups.io/g/lawsofform/topic/logical_graphs_formal/108456215 • https://groups.io/g/lawsofform/message/3766 AM: ❝GSB took J1: (a(a)) = as the first algebraic primitive and the second one is transposition so he only need only 2 primitives for the primary algebra. ❝Reflexion ((a))=a is proven without using Cancellation (( ))= . ❝In fact, he can prove cancellation from C1 reflexion. ❝Condensation ( )( ) = ( ) can be derived from C4 iteration. ❝So, his algebra is simpler from your Cactus Calculus.❞ Dear Arma, I had a feeling we've discussed this before, and probably in a lot more detail than I have time for at the moment (as I won't be online much for the next couple of weeks) so I hunted up the previous discussion — turns out it was on the old Yahoo Group — there's a copy of that below for whatever memory‑jogging it may be worth. To my way of thinking, what you say about reducing the primary arithmetic to the primary algebra shows a lack of appreciation for the fundamental nature of that distinction. Indeed, the recognition and clarification of that distinction is one of the most important upgrades Spencer Brown added to Peirce's initial systems of logical graphs. As far as the other score goes, the advantages of handling label changes and structure changes separately in one's syntactic operations is just one of those things I learned in the hard knocks way of programming theorem provers for logical graphs, and I all I can do is keep recommending it on that account. Regards, Jon