Aristotle (384-322 BC) was born Ἀριστοτέλης (Latin, Aristoteles) in the Greek colony of Stagira in the Chalcidice peninsula, not far from where the later monasteries would be on 
Mt. Athos. It was also not far from the Macedonian capital at Pella, where Aristotle's father came to be employed as a court physician. At 17, Aristotle went to Athens to study at the Academy of Plato, where he was so outstanding as a student that people began to call him the "Mind," νοῦς, of the Academy.
However, when Plato died in 347, Aristotle was passed over as his successor. Some nepotism may have been involved, since Plato's nephew was the next Scholarch, Σχολάρχης, the "ruler of the school." Aristotle decided to seek his fortune elsewhere. After some adventures, danger, and tragedy, this eventually took him back to Pella, where he got the job of tutoring the King's son, the boy who would be the future Alexander the Great. We have no idea what Aristotle taught him or what their relationship would be like, or what it would be like, if anything, in the future. Instead, although Alexander took along Aristotle's nephew Callisthenes on the invasion of Persia, Alexander ended up executing him (in 327), apparently over nothing more than disagreements.
Meanwhile Aristotle had returned to Athens in 335/34 and founded his own school, the Lyceum, Λύκειον, which was also in a grove, like the Academy, outside of Athens, but on the opposite side of the city from the Academy. Socrates mentions seeing Euthyphro at this grove, which means there may have been an athletic field, a gymnasium, γυμνάσιον ("place to be naked"), and perhaps a bath there. This is where Aristotle developed his mature thought, although none of it was left to us except in lecture notes, which had to be organized, edited, and published after his death -- with anomalies like the accidental title of the Metaphysics.
Of all the areas in Aristotle's philosophy, logic continues to involve issues of current interest. Indeed, modern and recent philosophers have often neglected to consider very basic issues of logic. This is one of the reasons for the importance of the Friesian School, which is rightly praised by Karl Popper for progress in these matters.
It is surprising how often we find things ignored or misunderstood that were clearly and decisively described by Aristotle. Thus, both Leibniz and the Logical Positivists thought that logic could be reduced to a mathematical and mechanical system such that all the problems of philosophy could simply be decided by "calculation," without the messy mediation of thought. Thank God! We don't need to think anymore!
What this would amount to became evident when the Positivists inherited and developed the forms of modern Symbolic Logic, which looked like it should fulfill the terms of what Leibniz had originally imagined.
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| Laurent de La Hyre (1606-1656), Allegory of Dialectic, 1650, Private Collection; one of the seven Liberal Arts |
Thus, the very definition of a valid deductive argument, which applies to Symbolic Logic fully as much as to Aristotle's logic, is that an argument is valid if and only if it is impossible for the premises to be true and the conclusion false. Thus, if the premises are true, the conclusion must be true. That's a big "if." A deductive argument, and the whole of Symbolic Logic, is only as good as its premises. But what makes the premises true? Aye, there's the rub.
This means that the maxim of computer programming, "Garbage in; garbage out," applies to all of logic as well. And this also means, as Robert Heinlein said, that if you give a philosopher enough paper, he can prove anything. Or, as Ayn Rand liked to say, "What are your premises?"
This did not trouble Leibniz, for whom a mathematicized logical system didn't exist, and who was entirely confident and complacent with the Rationalist certainty that the premises of his thought were all self-evident truths. This was still the heritage of Aristotle. But the collapse of Rationalism in the critiques of Hume and Kant left the Positivists in a difficult position. They could not appeal to self-evident truths except as principles of logic itself; and their supine reverence for science did not entirely blind them to the Problem of Induction, which had been weaponized by Hume. So what were they to do about their premises, their First Principles? Evasion seems to have been the answer. This was done either through confusion or dishonesty, which could be combined in a mash-up of self-deception and arrogance.
"Evasion," of course, meant that unjustified, unchallenged, and unexamined premises were smuggled into the arguments. With the Positivists, these were usually about logic, meaning, and language, so that a favorite dismissal by the Positivists of propositions, for instance, in ethics or metaphysics, was that they involved a "misuse of language." The basis or provenance of their authoritative and normative knowledge of language was glossed over. And since they had little real understanding of natural languages, their assertions about language were usually ill-informed and often absurd. When I tried taking a class in the Philosophy of Language at UCLA in 1968, and the professor said that the "language" he would address that quarter would be mathematics, I realized that I would derive nothing of value from the class. I wanted to ask the professor how you would ask "Where is the bathroom?" in the "language" of mathematics. Obviously, academic philosophers don't need to ask about bathrooms.
Even when Wittgenstein attacked many of the principles of Logical Positivism, he continued with the approach that his own special, unique, normative, and revelatory insights into the essence of language -- in a world, according to him, without essences -- enabled him to resolve all the questions of philosophy -- mainly to dismiss them, as the Positivists had, as meaningless non-questions. Answering one kind of Nihilism with another does not seem, to me, like a distinction that makes a difference. But a Nietzschean Nihilism continues to be the default axiology of the bien pensants of the modern academy.
So celebrated modern philosophers and their schools still don't know what to do about premises and First Principles. Rather than allow them their evasions, we can begin with Aristotle. 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" (ἀρχαί, principia prima) or "the first principles of demonstration" (principia prima demonstrationis).
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. Induction means simply counting examples of a type of thing or a phenomenon. Once you count enough, you get the feeling that everything must continue the characteristics that you have been counting. You generalize from individuals to universals.
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" (νοῦς, noûs). Thus, when your generalization is jusified, your "mind" makes a leap and arrives at a truth that is then self-evident. This ties the system of deduction and induction together.
A deductive system from first principles (like Euclidean geometry) is then what Aristotle calls "knowledge" (ἐπιστήμη, 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. That the Rationalists, on the basis of their own self-evident principles, came up with radically different systems, managed to refute the basis of their own kind of philosophy. The Dualism of Descartes, the Monism of Spinoza, and the Pluralism of Leibniz sank Rationalist from its own weight, without any need for external critics.
The Empiricists, like Locke, Berkeley, and Hume, thought that knowledge was mainly a matter of induction. But Hume weaponized the Problem of Induction by noting that no generalizations whatsoever are logically justified. 
In more modern terms, we never know when the individual will turn up to falsify any generalization. All around the world, swans are white. But when you get to Australia, there are black swans. Thus, the dreaded "black swan" is now the anomalous datum that sinks your merry little theory.
The Empiricist tradition thus culminated in Skepticism, Hume's conclusion that knowledge in the traditional sense does not exist. Nevertheless, Hume resorted to something rather like νοῦς himself. Commentators think that he "refuted" causality, since no enumeration of causes can establish the principle of causality; but Hume admitted that the mind "takes a step" from generalization to certainty that is not justified by reason. His only argument is that the uniform lack of counter-examples establishes a "habit" in the mind, which we then take for granted. This is simply part of human nature, and it rules out jumping from the roof rather than taking the stairs. Similarly, morality is established by custom and the proof of practice.
The Rationalists, in turn, were of course embarrassed that their systems, supposedly based on self-evident truths, nevertheless all contradicted each other. Symbolically, the separated branches of the arch, with deduction on one side and induction on the other, without the keystone of self-evidence, 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. Fries added that they were not known intuitively at all. Finally, Karl Popper resolves the regress of reasons, at least for scientific method, by substituting falsification for verification. But this also turns out to apply to Socratic Method.
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