The most important discovery in Future Logic is the formalization of induction by generalization and particularization (Part VI, chapters 50-59 and the appendix).
This work is crucial because it frees formal logic, and therefore epistemology, from the oversimplification of purely deductive approaches.
But to well understand this research, one ought to get acquainted with the sections concerning categorical modal propositions (Part II, chapters 11-17) and conditional modal propositions (Part IV, chapters 33-40).
These preliminary parts include, respectively, an inchoate formalization of the logic of change (i.e. of the dynamic aspects of classification, see ch. 17) and of certain modalities of causal logic (except for issues of volition).
It would be useful, to conclude this topic, to also read chapter 67.
The listed chapters would in themselves suffice, but it must be stated that Future Logic addresses the epistemological context, for I understand the term “logic” in its broader sense.
In this regard, I would like to also recommend reading of chapters 2, 4, 20, 31, 60, 61, 62, 64 and 66.
You will find a summary of the book in chapter 68.