Kant defines a priori knowledge as that kind of knowledge which is held independently of all experience, whereas empirical knowledge is possible only through experience. He further qualifies a priori with the adjective pure when the proposition in question contains no empirical elements, citing “every alteration has its cause” as a proposition which is a priori but not pure on the grounds that “alteration” is an empirical concept. Only a priori concepts, says Kant, can have properties like strict necessity and strict universality, since these properties could never be ascertained empirically. All mathematical propositions, for example, are a priori.
Similarly, Kant distinguishes between analytic and synthetic judgments, describing analytic judgments as those in which the subject overtly or covertly contains (or excludes) the predicate, such that the truth of the proposition is determined entirely by (the meanings of) the terms themselves. Synthetic judgments are those which are not analytic, meaning that the terms alone are insufficient to determine the truth of the proposition. As examples, Kant suggests, “all bodies are extended” as an analytic judgment, and “all bodies are heavy” as a synthetic judgment. Regardless of how compelling (or not) the examples are, the analytic/synthetic distinction itself was not controversial (or even novel) at the time.
It seems necessary that analytic judgments be a priori; once a judgment has been recognised as analytic, surely no reference to experience is needed, as the truth of the proposition can be determined simply by examining the terms. That being so, non a priori (ie empirical) knowledge is necessarily not analytic (ie synthetic) as a matter of logic. But these (apparent) implications do not preclude the possibility of knowledge which is both a priori and synthetic. Such knowledge, if it exists, would have the remarkable property that it grants us, without the need for experience, knowledge of truths which are not mere tautologies. When viewed in those terms, it seems almost spooky!
But do synthetic a priori judgments exist, and when (if at all) are we actually justified in calling them “knowledge”? Indeed, they do exist; it’s not unreasonable to say that all the really “interesting” judgments are exactly of this kind. “Every event has a cause” is an example – one that is immediately recognisable as the kind of judgment accepted as knowledge by the rationalists (although they failed to recognise it as synthetic). But the antithesis, “no event has a cause”, is also a synthetic a priori judgment: the fact that I can phrase it negatively without producing a contradictory or meaningless statement is sufficient to demonstrate that it is synthetic, and its claims to universality and necessity preclude it from being determined empirically.
How do we determine which (if either) of these can be considered knowledge? What are our grounds? Kant says that, whereas we can appeal to experience in the case of empirical judgments, we need some substitute source of appeal, “X”, in the case of synthetic a priori judgments, since ordinary experience can no longer help us. That the need for this mysterious “X” has not been recognised in the past, says Kant, was the cause of many long and fruitless speculative arguments and investigations.
Other examples of fields containing synthetic a priori judgments include pure mathematics and pure science. Pure science might raise an eyebrow at first, given as how it is firmly rooted in the empirical, but scientific laws like Newton’s laws of mechanics are synthetic a priori judgments: they make necessary and universal claims (thus a priori) which are more than simple logical imperatives (thus synthetic). And mathematics, although rich with logic and proof-by-contradiction (and thus apparently analytic, as supposed by Hume), is ultimately synthetic at its roots, and thus synthetic at its leaves (since it is not possible to arrive at an analytic result from a synthetic base, even when each step along the way is analytic). That mathematics is ultimately synthetic is by no means obvious, but Kant’s stance on the matter seems to have been vindicated with time; if my understanding is correct, Gödel’s incompleteness theorem has demonstrated that no fully satisfactory purely analytic account of mathematics can ever be forthcoming.
Similarly, all the important statements of metaphysics are synthetic a priori propositions. When viewed this way, the problem of metaphysics becomes one of determining how to make synthetic a priori judgments in that domain. Given the success of mathematics and natural science, which also deal with synthetic a priori judgments, there seems cause to hope for at least some semblance of real progress in the field when the appropriate rules of the game are recognised.
These “appropriate rules”, says Kant, must arise from reason itself, but not unbridled reason, free to speculate as it pleases. That, presumably, would be no better than a kind of natural science which operates speculatively from an armchair and never goes out of doors for a reality check. But metaphysics has no access to the great outdoors in the way that the natural sciences do (or else rationalism would have been less problematic). So the best “reality check” we can offer on reason is reason itself: reason scrutinising itself; a critique of pure reason.
Further: what is the relationship between the a priori and the noumenal realm of “things in themselves” that cannot be accessed by experience? here is an answer.
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