Chapter 8 is about proof and certainty. To prove an idea, one needs to link it back to perceived fact. The Objectivist term for this process of going back down the hierarchy to prove an idea is reduction. Contrary to contemporary notions, there is only one logic, not both one of discovery and one of proof. Instead, there are two different directions of motion along the same logical, hierarchical structure – derivation moves "up" from the perceptually given, while proof moves back "down" to the perceptually given.
"New knowledge can contradict old mistaken beliefs, but not old knowledge." He gives the example of when black swans were discovered in Australia. "The generalization "Swans are white" could not logically have warranted making the assertion: "There are no black swans anywhere in the world." That is not what was known at the earlier stage. The new knowledge is: "Swans are white, except in Australia where some are black." Thus, the end result is more knowledge, not less."
The three sources of cognitive errors are illogic, false premises, and incomplete information.
Knowledge and certainty are distinguishable concepts. Knowledge is differentiated from ignorance; certainty is differentiated from states that are less so. "Certainty" refers to cognitive status. Knowledge has both a metaphysical and epistemological component. "Fact" is purely metaphysical. Certainty is contextual.
Binswanger's formulation of the Law of Rationality is: In reaching conclusions, consider all the evidence and only the evidence.
There are sections on arbitrary ideas, the ad ignorantiam fallacy, and the burden of proof principle. He presents Ayn Rand's concept of objectivity. The final section is on the intrinsic-subjective-objective trichotomy. He illustrates it in regard to concepts. Most of that is covered in Chapter 3, on which I commented in #3 and #4 of this series.