Time and again Peirce refers to his logic of relatives as the means necessary to understand the more complex issues in his theory of inquiry and theory of signs. I find this to be good advice.
The best antidote for confusion about triadic sign relations and the three basic modes of inference can be found in the study of Peirce’s early papers on the logic of relatives and the logic of science. His first expeditions, for all their rough and exploratory character, perhaps even because of it, give far more concrete examples of relations in general and triadic sign relations in particular, plus a better idea of actual practice in the ways of inquiry, than the often detached abstractions of his later speculations and summations.
From what I’ve seen through many years of watching people struggle with Peirce, it is almost impossible to get what Peirce is talking about in his later work without getting a foothold on the concrete foundation he laid down at the outset of his work.
I think a close reading of Peirce’s 1870 “Logic of Relatives” could be extremely beneficial in understanding and applying Peirce’s ideas to realistic and significant open problems. It’s where I began my own acquaintance with Peirce’s work and I’ve put my notes on the foothills of the 1870 paper on the web several times, the current best version on the InterSciWiki, here: