Except for outright Skeptics, Aristotle's solution to the Problem of First Principles, that such propositions are known to be true because they are self-evident, endured well into Modern Philosophy. Then, when all the Rationalists, like Descartes, Spinoza, and Leibniz, appealed to self-evidence and then all came up with radically different theories, it should have become clear that this was not a good enough procedure to adjudicate the conflicting claims. This awkward situation was then blown apart by Hume, under whose skeptical examination, reviving the critique of al-Ghazâlî, even the principle of causality crumbled.
Kant does not directly pose the Problem of First Priniciples, and the form of his approach tends to obscure it. Thus, the "Transcendental Logic" in the Critiqiue of Pure Reason is divided into the "Transcendental Analytic" and the "Transcendental Dialectic." The "Dialectic" is concerned with the fallacies produced when metaphysics is extended beyond possible experience. The "Analytic," about secure metaphysics, is divided into the "Analytic of Concepts" and the "Analytic of Principles." "Principles" would be Principia in Latin, i.e. "beginnings," "first things," "first principles," where now in English, thanks to the drift in the meaning of "principle," the term must be reduplicated with an etymologically redundant "first." Kant, however, is here writing in German, and in place of Principia we have Grundsätze (singular Grundsatz, "principle," "axiom" -- literally "ground sentence"). The examination of the Grundsätze, however, is deferred until after and "Analytic of Concepts." Thus, were the Problem of First Principles to be raised, it seems like that would come after an examination of concepts. Since it is not raised at all, one is left with the impression that it has somehow, along the way, actually already been dealt with. It has.
The peculiarity of Kant's approach, from an Aristotelian (or Friesian) point of view, is not idiosyncratic. Kant approaches the matter as he does because he is responding to Hume, and one of Hume's intitial challenges is about the origin of "ideas." While the Problem of First Principles is about the justification of propositions, Hume's Empiricist approach goes back to asking about the legitimacy of the very concepts, of which the propositions are constituted, in the first place. The Rationalists never worried too much about that. For Descartes, any notion that could be conceived "clearly and distinctly" could be used without hesitation or doubt, a procedure familiar and unobjectionable in mathematics. It was the Empiricists who started demanding certificates of authenticity, since they wanted to trace all knowledge back to experience. Locke was not aware, so much as Berkeley and Hume, that not everything familiar from traditional philosophy (or even mathematics) was going to be so traceable; and Berkeley's pious rejection of "material substance" lit a skeptical fuse whose detonation would shake much of subsequent philosophy through Hume, thanks in great measure to Kant's appreciation of the importance of the issue.
Thus, Kant begins, like Hume, asking about the legitimacy of concepts. However, the traditional Problem has already insensibly been brought up; for in his critique of the concept of cause and effect, Hume did question the principle of causality, a proposition, and the way in which he expressed the defect of such a principle uncovered a point to Kant, which he dealt with back in the Introduction to the Critique, not in the "Transcendental Logic" at all. Hume had decided that the lack of certainty for cause and effect was because of the nature of the relationship of the two events, or of the subject and the predicate, in a proposition. In An Enquiry Concerning Human Understanding, Hume made a distinction about how subject and predicate could be related:
All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of Fact. Of the first kind are the sciences of Geometry, Algebra, and Arithmetic; and in short, every affirmation which is either intuitively or demonstratively certain [note: these are Locke's categories]. That the square of the hypothenuse is equal to the square of the two sides, is a proposition which expresses a relation between these figures. That three times five is equal to the half of thirty, expresses a relation between these numbers. Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence.
Matters of fact, which are the second objects of human reason, are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing. The contrary of every matter of fact is still possible; because it can never imply a contradiction, and is conceived by the mind with the same facility and distinctness, as if ever so conformable to reality. That the sun will not rise to-morrow is no less intelligible a proposition, and implies no more contradiction than the affirmation, that it will rise. We should in vain, therefore, attempt to demonstrate its falsehood. Were it demonstratively false, it would imply a contradiction, and could never be distinctly conceived by the mind. [Enquiries, Selby-Bigge edition, Oxford, 1902, 1972, pp.25-26]
Both paragraphs warrant quoting in full. The first now would seem properly more a matter of embarrassment than anything else. Whatever Hume expected from intuition or demonstration, it would be hard to find a mathematician today who would agree that "the truths demonstrated by Euclid would for ever retain their certainty and evidence." If Hume's fame rests on this point, there would be little to recommend it. The second paragraph, however, redeems the impression by giving us a logical criterion to distinguish between truths that are "relations of ideas" and those that are "matters of fact": A matter of fact can be denied without contradiction.
This was the immediate inspiration to Kant, who can have asked himself how something "demonstratively false" would "imply a contradiction." A contradiction means something of the form "A and not-A." If a proposition expressing a matter of fact can be denied without contradiction, then the subject and the predicate of such a proposition cannot contain anything in common, otherwise the item would turn up posited in the subject but negated in the predicate of the denial. On the other hand, a proposition that cannot be denied without contradiction must contain something in the predicate that is already in the subject, so that the item does turn up posited in the subject but negated in the predicate of the denial. This struck Kant as important enough that, like Hume, he founded a whole critique on it, and also produced some more convenient and expressive terminology. Propositions true by "relations of ideas" are now analytic ("taking apart"), while propositions not so founded are synthetic ("putting together").
This clarified distinction Kant could then turn on Hume's own examples of "relations of ideas." Can geometry be denied without contradiction? Kant did not see that the predicates of the axioms of geometry contained any meaning already expressed in the subjects. They were synthetic. They could be denied without contradiction. Geometry would thus not have an intuitive self-evidence or demonstrative certainty that Hume claimed for it. Kant still thought that Euclid, indeed, would have certainty, but the ground of certainty would have to located elsewhere. Nevertheless, Kant is rarely credited, and Hume rarely faulted, for their views of the logic of the axioms of geometry. If the axioms of Euclid can be denied without contradiction, this means that systems of non-Euclidean geometry are logically possible and can be constructed without contradiction. But it is not uncommon to see the claim that Kant actually denied this, and it is Kant, not Hume, who is typically belabored for implicitly prohibiting the development of non-Euclidean system. This distortion can only come from confusion and bias, a confusion about the meaning of "synthetic" (even in Hume's corresponding category), and a bias that the Analytic tradition has for British Empiricism, by which the glaring falsehood of Hume's statements is ignored and Kant's true and significant discovery misrepresented. This curious and reprehensible turn is considered in detail elsewhere.
Kant, as it happens, also did not see how arithmetic could be analytic. In his own example of "7 + 5 = 12" (p. B-15), if "7 + 5" is understood as the subject, and "12" as the predicate, then the concept or meaning of "12" does not occur in the subject. This was rather harder to swallow than the point about geometry, for it seems rather "intuitively" certain that "7 + 5 = 12" cannot be denied without contradiction. Kant must have missed something. Hope for demonstrating the analytic nature of arithmetic came with the development of propositional logic, since a proposition like "P or not P" clearly cannot be denied without contradiction, but it is not in a subject-predicate form. Still, "P or not P" is still clearly about two identical things, the P's, and "7 + 5 = 12" is more complicated than this. But, if "7 + 5 = 12" could be derived directly from logic, without substantive axioms like in geometry, then its analytic nature would be certain. In their Principia Mathematica (1910-1913), Russell and Whitehead and, in the Tractatus, Wittgenstein thought that they could indeed derive arithmetic from logic. Their demonstrations, however, were flawed, and it turned out that substantive axioms were necessary, just like in geometry. The axioms are now those of axiomatic Set Theory, and it is Set Theory that concerns the foundations of arithmetic. Kant turned out to be right again, though, curiously, he is again rarely credited for this.
Kant's discovery, however, can be trivialized if it turns out that there are simply no analytic propositions at all. This task was undertaken by Willard Van Orman Quine ("Two Dogmas of Empiricism," 1950). The approach, simply enough, was Nominalistic. If we say that "red is a color" is an analytic proposition, where is "color" in "red"? I don't see it. If we say that the meaning "color" is in the meaning of "red," where are these "meaning" things? I don't see them. Thus, if language consists of words but not abstract meanings, then we don't have to worry about one meaning containing another. "Red is a color" is just a convention of our language, which is even what we can say about "P or not P." Besides the generally failings of Nominalism, Quine's particular critique is well refuted by Jerrold Katz.
In Kant there is little left in the category of "analytic." Definitions and truths of logic are going to be about it; and the definitions themselves will be suspect when the concepts defined may or may not be legitimate. The meaning within a concept must also in some sense be "put together," and the ground of this will raise the same questions as the ground of synthetic propositions. Thus, Saul Kripke began to speak of "analytic a posteriori" propositions, when the meanings in the subject are themselves united on only a posteriori grounds, i.e. the basis of experience. Indeed, dictionary definitions of natural language words are prima facie of conventional usage, e.g. how a pot is different from a pan, and the meaning of any words can be simply stipulated for some appropriate purpose, e.g. a "designated hitter" can go to bat for some particular member of a baseball team (usually the pitcher), without otherwise replacing him in other play. Thus, a big fight over the existence of analytic propositions doesn't in the end make that much difference. Synthetic propositions are the key anyway, as they were if Kant wanted to answer Hume's critique of causality.
For, indeed, outside of an axiomatized logic itself, the First Principles of Demonstration will be synthetic. However Kant can explain the truth of non-empirical synthetic propositions, i.e. those that are a priori instead of a posteriori, that will be his answer to the Problem of First Principles. They are clearly now, after Hume, not going to be self-evident. Yet Hume himself is often poorly understood. While it is common to say that Hume denied the existence of synthetic a priori propositions, there is some question about whether he actually does. He says that the relationship of cause and effect is not discovered or known by any reasonings a priori, but that is not the same thing. A synthetic a priori proposition is not known from any reasonings. In fact, Hume does not see that the relationship of cause and effect is discovered or known from anything, since it is not justified by experience, in which there is no necessary connection between cause and effect, and there is in fact nothing in the cause to even suggest the effect, much less than the effect must follow. Hume's famous explanation was a psychological one, that we become accustomed to the association of certain events ("causes") with others ("effects"); but this, obviously, carries no weight whatsoever about the nature of things, which is what makes Hume, very properly, a Sketpic.
At the same time, Hume had no doubts whatsoever of the necessity of cause and effect. This is where he is commonly misrepresented. People assume that because he was a Skeptic, then he must have thought it possible for causes to occur without effects, i.e. for the principle of causality to be contradicted in actuality. He never had any such expectation, and in fact he ruled out a priori, not only miracles, but also chance and free will just because they would violate (a very deterministic) causality. Confusion over this occurs because people do not appreciate that Hume as an "Academic" Skeptic, holding that lack of knowledge (the meaning of "Skepticism") does not rule out "reasonable" beliefs. Causality is a "reasonable" belief because, as Hume says, "All reasonings concerning matter of fact seem to be founded on the relation of Cause and Effect" [Enquiry, op. cit., p. 26]. So without it, we would have no basis of reasoning in daily life. Thus, Hume says:
Nor need we fear that this philosophy, while it endeavors to limit our enquiries to common life, should ever undermine the reasonings of common life, and carry its doubts so far as to destroy all action, as well as speculation. Nature will always maintain her rights, and prevail in the end over any abstract reasoning whatsoever. Though we should conclude, for instance, as in the foregoing section, that, in all reasonings from experience, there is a step taken by the mind which is not supported by any argument or process of the understanding [i.e. from cause to effect]; there is no danger that these reasonings, on which almost all knowledge depends, will ever be affected by such a discovery. [ibid., p. 41]
Kant therefore understood that Hume's problem was not with the quid facti, that there were causes and effects, and necessary connection, but with the quid juris, the epistemic justification of the principle. While some philosophers spent much of the 20th Century congratulating Hume for having discovered that causality might not exist, they never seem to have noticed that he explicitly denied having done anything of the sort. Kant already knew the type, who "were ever taking for granted that which he doubted, and demonstrating with zeal and often with impudence that which he never thought of doubting..." [Prolegomena to Any Future Metaphysics, p. 259, Lewis White Beck translations, Bobbs-Merrill, 1950, p.6].
Kant's solution to the quid juris in the Critique of Pure Reason was the argument of the "Transcendental Deduction" (in the "Analytic of Concepts") that concepts like causality are "conditions of the possibility of experience," because they are the rules by which perception and experience are united into a single consciousness, through a mental activity called "synthesis." Once the existence of consciousness is conceded (which not everyone, e.g. behaviorists, might be willing to do), then whatever is necessary for the existence of consciousness must be conceded.
This is a strong argument and, decisive or not, is heuristically of great value, especially when we untangle it from the earlier views of perception in the Critique. However, it suffers from a couple of serious drawbacks. One is that, like Hume's own explanation, it is a psychological approach that does not necessarily tell us anything about objects, i.e. consciousness may be united in a way that is irrelevant to external things. Kant seemed to recognize this himself when he said that none of this gives us any knowledge of things-in-themselves. This problem was never properly sorted out by Kant, and is considered independently in "Ontological Undecidabilty".
The second drawback of Kant's argument is that it would only work, indeed, for the "conditions of the possibility of experience," and not for any other matters which might seem to involve synthetic a priori propositions. Hume himself was just as concerned about morality as about causality, and found himself in the same Skeptical position in both matters. The only comparable thing that Kant can do for morality, however, would be to employ a principle of the "conditions of the possibility of morality." But this would require conceding that morality exists, which is something that a very large number of people in the 20th century, far beyond behaviorists, would not be willing to do. Nor does it make one a Kantian merely to vaguely appeal to human "rationality" (e.g. John Rawls) as a basis for morality, since this really just begs the question of justification -- besides violating Hume's famous observation that propositions of obligation ("ought," imperatives) cannot be logically derived from propositions of fact ("is," indicatives).
Keeping in mind that First Principles cannot be proven, and that synthetic propositions can be denied without contradiction, the conspicious historical alternatives seem to be to deny one or the other. Hegel denied the first, by taking the equivalent of Kant's Transcendental Deduction as itself a part of metaphysics and a proof, by means of novel principles of "dialectical" logic, of moral and metaphysical truths. To an extent, Hegel may also have denied the second, as Leibniz certainly did, treating any moral or metaphysical truth as analytic, if only from the point of view of divine omniscience. Either such move, however, cannot escape the original embarrassments of Rationalism, or avoid the devastation inflicted by the criticisms made by Hume and Kant.
Less conspicous historically was Jakob Fries, who could accept the proper meanings of "First Principle" and of synthetic propositions. The Friesian theories of deduction and of non-intuitive immediate knowledge make it possible to preserve the advances of Hume and Kant without falling back into Rationalism or heading for the Nihilism (so different from Hume's Skepticism), relativism, scientism, pragmatism, etc., so conspicuous in the 20th century. Later, Karl Popper proposed a special solution for the Problem in that science, by using falsification, does not need to worry about a positive justification of First Principles at all. This enables scientific progress to heedlessly continue, as it has, regardless of the status of any philosophical solution.
Thus, Kant gave us the real elements of the solution of the Problem of First Principles, even though he could not complete and seal the matter himself. Indeed, no one can hope to do that, even as new elements and new understanding of the solution emerge over time.