Comments I made elsewhere about computer science and (anti-)social networks have a connection with the work in progress on this thread, so it may steal a march to append them here.
I have been interested for a long time now in using graphs to do logic. For that you need more than positive links — negative relations are more generative than positive relations. The logical situation is analogous to social networks where people can “unlike” or “¬like” other people, or website networks where the information at one node may contradict the information at another node. In my pursuits it turns out that particular species of graph-theoretic “cacti” are much more useful than the garden variety trees and unsigned graphs.
For what it’s worth, here is my exposition of “painted cacti” and their application to propositional calculus.
A painted cactus is a rooted cactus with any number of symbols from a finite alphabet attached to each node. In their ordinary logical interpretations these symbols (“paints”) stand for boolean variables.
Triangles are interesting in computational contexts because they arise in case-breakdown expressions. In one of the common interpretations of cactus graphs, a rooted triangular lobe says the values of the two non-root nodes are logically inequivalent.
- Logic Syllabus
- Logical Graphs • (1) • (2)
- Futures Of Logical Graphs
- Propositional Equation Reasoning Systems
- Differential Propositional Calculus • Part 1 • Part 2