The chart below graphically represents Aristotle's view of how knowledge is produced.
Our attempt to justify our beliefs logically by giving reasons results in the "regress of reasons." Since any reason can be further challenged, the regress of reasons threatens to be an infinite regress. However, since this is impossible, there must be reasons for which there do not need to be further reasons: Reasons which do not need to be proven. By definition, these are "first principles" or "the first principles of demonstration." The "Problem of First Principles" arises when we ask Why such reasons would not need to be proven. Aristotle's answer was that first principles do not need to be proven because they are self-evident, i.e. they are known to be true simply by understanding them.
But, Aristotle thinks that knowledge begins with experience. We get to first principles through induction. But there is no certainty to the generalizations of induction. The "Problem of Induction" is the question How we know when we have examined enough individual cases to make an inductive generalization. Usually we can't know.
Thus, to get from the uncertainty of inductive generalizations to the certainty of self-evident first principles, there must be an intuitive "leap," through what Aristotle calls "Mind." This ties the system together. A deductive system from first principles (like Euclidean geometry) is then what Aristotle calls "knowledge" ("epistemê" in Greek or "scientia" in Latin). The Rationalists, such as Descartes, Spinoza, and Leibniz, later thought that the part of the system with self-evident first principles and deduction was all that was necessary to do philosophy.
Self-evidence breaks down as a solution to the Problem of First Principles because there is no way to resolve disputes about whether something is self-evident or not. The domain of the self-evident is drastically reduced by Hume and Kant. The Empiricists, like Locke, Berkeley, and Hume, thought that knowledge was mainly a matter of eduction. However, Hume sharpened the Problem of Induction by noting that no generalizations whatsoever are logically justified. The Empiricist tradition thus culminated in Skepticism, Hume's conclusion that knowledge in the traditional sense does not exist. The Rationalists, in turn, were embarrassed that their systems, suposedly based on self-evident truths, nevertheless all contradicted each other. Symbolically, the separated branches of the arch obviously are unstable and cannot stand independently.
Kant proposed a different solution to the Problem of First Principles: synthetic a priori propositions are first principles of demonstration but are not self-evident. Finally, Karl Popper resolves the regress of reasons, at least for scientific method, by substituting falsification for verification.
The Friesian Trilemma
The Münchhausen Trilemma
In Defense of Bramantip
The Foundations of Value, Logical Issues
History of Philosophy