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There are many points of view on these two types of knowledge, and their relationship is one of the oldest problems in modern philosophy.
The terms a priori and a posteriori are primarily used as adjectives to modify the noun "knowledge", or taken to be compound nouns that refer to types of knowledge (for example, "a priori knowledge"). However, "a priori" is sometimes used as an adjective to modify other nouns, such as "truth". Additionally, philosophers often modify this use. For example, "apriority" and "aprioricity" are sometimes used as nouns to refer (approximately) to the quality of being "a priori".
Although definitions and use of the terms have varied in the history of philosophy, they have consistently labelled two separate epistemological notions. See also the related distinctions: deductive/inductive, analytic/synthetic, necessary/contingent.
The intuitive distinction between a priori and a posteriori knowledge (or justification) is best seen in examples.
For another comparison, the mathematical theory of everything model for our universe is a posteriori knowledge; but the purely mathematical manipulation of those same equations is justified a priori knowledge.
Several philosophers reacting to Kant sought to explain a priori knowledge without appealing to, as Paul Boghossian (MD) explains, "a special faculty...that has never been described in satisfactory terms." One theory, popular among the logical positivists of the early twentieth century, is what Boghossian calls the "analytic explanation of the a priori." The distinction between analytic and synthetic propositions was first introduced by Kant. While Kant's original distinction was primarily drawn in terms of conceptual containment, the contemporary version of the distinction primarily involves, as the American philosopher W. V. O. Quine put it, the notions of "true by virtue of meanings and independently of fact." Analytic propositions are thought to be true in virtue of their meaning alone, while a priori synthetic propositions are thought to be true in virtue of their meaning and certain facts about the world. According to the analytic explanation of the a priori, all a priori knowledge is analytic; so a priori knowledge need not require a special faculty of pure intuition, since it can be accounted for simply by one's ability to understand the meaning of the proposition in question. In short, proponents of this explanation claimed to have reduced a dubious metaphysical faculty of pure reason to a legitimate linguistic notion of analyticity.
However, the analytic explanation of a priori knowledge has undergone several criticisms. Most notably, Quine argued that the analytic-synthetic distinction is illegitimate. Quine states: "But for all its a priori reasonableness, a boundary between analytic and synthetic statements simply has not been drawn. That there is such a distinction to be drawn at all is an unempirical dogma of empiricists, a metaphysical article of faith." While the soundness of Quine's critique is highly disputed, it had a powerful effect on the project of explaining the a priori in terms of the analytic.
The metaphysical distinction between necessary and contingent truths has also been related to a priori and a posteriori knowledge. A proposition that is necessarily true is one whose negation is self-contradictory (thus, it is said to be true in every possible world). Consider the proposition that all bachelors are unmarried. Its negation, the proposition that some bachelors are married, is incoherent, because the concept of being unmarried (or the meaning of the word "unmarried") is part of the concept of being a bachelor (or part of the definition of the word "bachelor"). To the extent that contradictions are impossible, self-contradictory propositions are necessarily false, because it is impossible for them to be true. Thus, the negation of a self-contradictory proposition is supposed to be necessarily true. By contrast, a proposition that is contingently true is one whose negation is not self-contradictory (thus, it is said that it is not true in every possible world). As Jason Baehr states, it seems plausible that all necessary propositions are known a priori, because "[s]ense experience can tell us only about the actual world and hence about what is the case; it can say nothing about what must or must not be the case."
Following Kant, some philosophers have considered the relationship between aprioricity, analyticity, and necessity to be extremely close. According to Jerry Fodor, "Positivism, in particular, took it for granted that a priori truths must be necessary...." However, since Kant, the distinction between analytic and synthetic propositions had slightly changed. Analytic propositions were largely taken to be "true by virtue of meanings and independently of fact", while synthetic propositions were not—one must conduct some sort of empirical investigation, looking to the world, to determine the truth-value of synthetic propositions.
Aprioricity, analyticity, and necessity have since been more clearly separated from each other. The American philosopher Saul Kripke (1972), for example, provided strong arguments against this position. Kripke argued that there are necessary a posteriori truths, such as the proposition that water is H2O (if it is true). According to Kripke, this statement is necessarily true (since water and H2O are the same thing, they are identical in every possible world, and truths of identity are logically necessary) and a posteriori (since it is known only through empirical investigation). Following such considerations of Kripke and others (such as Hilary Putnam), philosophers tend to distinguish more clearly the notion of aprioricity from that of necessity and analyticity.
Kripke's definitions of these terms, however, diverge in subtle ways from those of Kant. Taking these differences into account, Kripke's controversial analysis of naming as contingent and a priori would best fit into Kant's epistemological framework by calling it "analytic a posteriori".[nb 1]
Thus, the relationship between aprioricity, necessity, and analyticity is not easy to discern. However, most philosophers at least seem to agree that while the various distinctions may overlap, the notions are clearly not identical: the a priori/a posteriori distinction is epistemological, the analytic/synthetic distinction is linguistic, and the necessary/contingent distinction is metaphysical.
The phrases "a priori" and "a posteriori" are Latin for "from what comes before" and "from what comes later" (or, less literally, "[from first principles, but] before experience" and "after experience"). They appear in Latin translations of Euclid's Elements, a work widely considered during the early European modern period as the model for precise thinking.
An early philosophical use of what might be considered a notion of a priori knowledge (though not called by that name) is Plato's theory of recollection, related in the dialogue Meno (380 B.C.), according to which something like a priori knowledge is knowledge inherent, intrinsic in the human mind.
Albert of Saxony, a 14th-century logician wrote on the both a priori and a posteriori. George Berkeley, the Irish divine and philosopher outlined the distinction in A Treatise Concerning the Principles of Human Knowledge of 1710, though the terms were already well known by that time.
The eighteenth-century German philosopher Immanuel Kant (1781) advocated a blend of rationalist and empiricist theories. Kant states, "although all our knowledge begins with experience, it does not follow that it arises from experience" According to Kant, a priori knowledge is transcendental, or based on the form of all possible experience, while a posteriori knowledge is empirical, based on the content of experience. Kant states, "... it is quite possible that our empirical knowledge is a compound of that which we receive through impressions, and that which the faculty of cognition supplies from itself (sensuous impressions giving merely the occasion)." Thus, unlike the empiricists, Kant thinks that a priori knowledge is independent of the content of experience; moreover, unlike the rationalists, Kant thinks that a priori knowledge, in its pure form, that is without the admixture of any empirical content, is knowledge limited to the deduction of the conditions of possible experience. These a priori, or transcendental conditions, are seated in one's cognitive faculties, and are not provided by experience in general or any experience in particular. Kant nominated and explored the possibility of a transcendental logic with which to consider the deduction of the a priori in its pure form. Concepts such as time and cause are counted among the list of pure a priori forms. Kant reasoned that the pure a priori forms are established via his transcendental aesthetic and transcendental logic. He claimed that the human subject would not have the kind of experience that it has were these a priori forms not in some way constitutive of him as a human subject. For instance, he would not experience the world as an orderly, rule-governed place unless time and cause were operative in his cognitive faculties. The claim is more formally known as Kant's transcendental deduction and it is the central argument of his major work, the Critique of Pure Reason. The transcendental deduction does not avoid the fact or objectivity of time and cause, but does, in its consideration of a possible logic of the a priori, attempt to make the case for the fact of subjectivity, what constitutes subjectivity and what relation it holds with objectivity and the empirical.
After Kant's death, a number of philosophers saw themselves as correcting and expanding his philosophy, leading to the various forms of German Idealism. One of these philosophers was Johann Fichte. His student (and critic), Arthur Schopenhauer, accused him of rejecting the distinction between a priori and a posteriori knowledge:
...Fichte who, because the thing-in-itself had just been discredited, at once prepared a system without any thing-in-itself. Consequently, he rejected the assumption of anything that was not through and through merely our representation, and therefore let the knowing subject be all in all or at any rate produce everything from its own resources. For this purpose, he at once did away with the essential and most meritorious part of the Kantian doctrine, the distinction between a priori and a posteriori and thus that between the phenomenon and the thing-in-itself. For he declared everything to be a priori, naturally without any evidence for such a monstrous assertion; instead of these, he gave sophisms and even crazy sham demonstrations whose absurdity was concealed under the mask of profundity and of the incomprehensibility ostensibly arising therefrom. Moreover, he appealed boldly and openly to intellectual intuition, that is, really to inspiration.—Schopenhauer, Parerga and Paralipomena, Vol. I, §13