Notes on Kant’s Introduction to The Critique of Pure Reason

Hume held that it was impossible for a priori synthetic knowledge to be obtained, this of which makes it impossible for there to be a priori knowledge about the world. The goal of Kant’s Critique is thus an exposition to dispose of this, answering the question, “How are synthetic judgements a priori possible?” (B 19) Thus allowing for a (sort of) synthesis of both the traditional rationalism and empiricism. This essay will answer the question of what a priori vs a posteriori cognitions are; and the bilateral question of what analytic vs synthetic judgements are, thus allowing the concepts to be coagulated.

A Priori vs A Posteriori Cognitions

To begin, we shall distinguish between a priori and empirical knowledge. An a priori cognition allows for knowledge that is “independent of all experience, and even of all impressions of the sense” (B 2). Whereas an a posteriori cognition allows for knowledge that has its roots in empirical reality, obtained through experience (B 2). This however already runs into a problem, with “The main question arising from this definition [being] what constitutes ‘independence’ from experience.”¹ For, one could say that one knows a priori, without any experience, that a house would collapse without any foundation, by virtue of the fact that “bodies are heavy and hence fall if their support is taken away” (B 2), however, Kant does note that “one could not have known this entirely a priori” (B 2), as such information would “have [to have] become known to him through experience” (B 2). Thus, “a priori cognitions [are] not those that occur independently of this or that experience, but rather [are] those that occur absolutely independently of all experience” (B 2–3). Leading Kant to distinguish between pure and non-pure a priori cognitions, the pure cognitions of which contain nothing empirical. Thus making the former proposition regarding the house collapsing due to its lack of foundations not a pure a priori cognition, as it necessitates drawing a concept from experience. Thankfully, “Such indirect dependence on experience does not compromise the knowledge’s a priori status because we can ‘separate’ the a priori use of the faculty from its empirical use (B 1–2).”² Following from this, Kant can assert that although these concepts are learned through experience, they are still a priori: take, for instance, “Every alteration has a cause” (B 3); “The quantity of matter is conserved (B 21n); “Bachelors are unmarried.”³

Now we are to begin looking at how “to establish criteria for identifying a priori cognitions and deploy them to show that we have a priori knowledge.”⁴

“Experience teaches us, to be sure, that something is constituted thus and so, but not that it could not be otherwise. First, then, if a proposition is thought along with its necessity, it is an a priori judgement; … Second: Experience never gives its judgements true or strict but only assumed and comparative universality (through induction), so properly it must be said: as far as we have yet perceived, there is no exception … Necessity and strict universality are therefore secure indications of an a priori cognition” (B 3–4).

Now, there are already problems in the second way of defining a priori, as John Divers points out: “… some universal generalizations are clearly a posteriori (‘All ravens are black’); and some a priori truths are not universal (‘That man is not my wife,’ ‘There is a natural number less than 2,’ or arguably, ‘7 + 5=12’).”⁵ However, if you were to grant that they are biconditional then you must throw out the argument regarding ‘All ravens are black,’ as ravens being black is not thought along with its necessity. The argument ‘That man is not my wife’ can be thrown out as well, as it would be a performative contradiction for a man who does have a wife to state that about his wife, therefore it is not universal. As for the arithmetic examples, I don’t even know where you’d begin to affirm those as criticisms. Of course, there is another route you could take in this, which would make those two sufficient conditions not biconditional, in which case you would have to grant that the criticisms above are true. Divers solves for this by weakening Kant’s thesis,

“he focuses on apriority as applied to warrants rather than propositions, and limits himself to a sufficient condition only. Universality and explicit necessity turn out to be criteria for apriority in the sense of providing sufficient condition for there being at least some important a priori element in the best kinds of warrant for the claim.”⁶

From here, Kant points out that if we stick merely to these concepts of a priori and a posteriori then metaphysics turns out to be “nothing more than illuminations or clarifications of that which is already thought” (B 9). This of which leads us to the aforementioned bilateral question of analytic vs synthetic judgements.

Analytic vs Synthetic Judgements

This is where Kant’s innovation of the analytic/synthetic distinction comes out in full force. Kant from here discusses how he thinks analytic and synthetic judgements are to function,

“In all judgements in which the relation of a subject to the predicate is thought… this relation is possible in two different ways. Either the predicate B belongs to the subject A as something that is (covertly) contained in this concept A; or B lies entirely outside the concept A, though to be sure it stands in connection with it. In the first case I call the judgement analytic, in the second synthetic. Analytic judgements are thus those in which the connection of the predicate is thought through identity, but those in which this connection is thought without identity are to be called synthetic judgements. One could also call the former judgements of clarification, and the latter judgements of amplification, since through the predicate the former do not add anything to the concept of the subject, but only break it up by means of analysis into its component concepts, which were already thought in it (though confusedly); while the latter on the contrary add to the concept of the subject a predicate that was not thought in it at all, and could not have been extracted from it through any analysis” (A 6–7/B 10).

Although this sounds obvious, this has deep implications. For instance, this has the implication that we can have knowledge about the world (a priori synthetic), contrary to what Hume believed. Notice that I am specifically writing ‘about’, I’m not saying Hume believed that we can’t have knowledge of the world, rather he believed we can’t have knowledge about it.

I digress, an example of an analytic judgement would be, “All bodies are extended,” for the predicate is expressing something already contained in the subject. The rationalists of Kant’s time (such as Leibniz) also believed that mathematics was analytic, for, <7> added together with <5> must always equal <12>… However, Kant later goes on to argue mathematics is synthetic (as I will explain soon).

As for an example of a synthetic judgement, it would be, “All bodies are heavy,” for the concept of weight is not already contained in the concept of a body but is connected. Whereas extension (as noted earlier when talking about analytic judgements) is necessitated for the thought of a body. Kant later goes on to say that, “(1) All empirical judgements are synthetic (A 7/B 11); (2) Mathematics is synthetic (B 14); (3) The a priori parts of natural science are synthetic (B 17); and finally, (4) Metaphysical claims, if sustained, must likewise count as synthetic (B 18).”⁷ Kant would say mathematics belongs to this group of things (synthetic) because in the concept of “7 + 5 = 12” all that’s thought is the combination of <7> and <5>, not the particular number they combine into.

Unfortunately though, there are problems arisen already with these definitions. Seemingly, as shown in the quote block above, Kant gives three different definitions for analyticity (unknowingly)! (1) Analyticity “as concept containment: ‘Either the predicate B belongs to the subject A as something … contained in this concept A; or B lies entirely outside the concept A …’ (A 6–7/B 10).”⁸ (2) Analyticity as “thought through identity”, or in otherwords, the principle of contradiction. This one seems to be the one Kant intended, as Kant points out in the Prolegomena that, “All analytic judgements depend wholly on the principle of contradiction” (267)… From this — it is seeming to me — Kant intended that you could deduce all three of these definitions from this one. This notion of three definitions, I claim, is trivial, Kant simply deducing the first and third definition from the second definition shown. (3) Analyticity as “‘merely explicative,’ rather than ‘ampliative’” (B 11).⁹ R. Lanier Anderson makes the case that concept containment (the first definition) precedes the second and third, however, like I have remarked earlier, it is generally trivial. Thus I will — for this essay and future essays unless stated otherwise — equivalize definition one and two (making them biconditional), disregarding three (as it seems to run into questionable areas, especially if not made interrelated with the two prior).

Now that we have established analyticity above, we can get deeper into mathematical syntheticity. For you must agree with me that the lack of explanation required for mathematical syntheticity is apparent. If we are to claim that metaphysics is a science (metaphysics being a priori synthetic as aforementioned), and is of equal level to that of mathematics, you need quite the strong argument. Thankfully, Anderson does a beautiful job of explaining it,

“… that connection is constrained by reciprocity; if B is contained in A, then A must fall under B, and further, it must contain whatever B contains and exclude whatever B excludes, lest their contents and extensions come apart. But even elementary arithmetic truth requires contents and extensions to part company in just this way. Consider Kant’s example, “7 + 5 = 12”: if <12> is supposed to be contained in the sum concept, <7 + 5>, then <7 + 5> must exclude whatever <12> does, including, presumably, <5> and <7> (since 12 ≠ 5 and 12 ≠ 7). But now, since it permits only the one type of affirmative connection, containment analyticity affords no way of explaining the relation <7 + 5> bears to <7> and <5>, and thus it fails to express the relation among the three numbers that is essential to the content of the arithmetic proposition.”¹⁰

Simple, right? Unfortunately, Kant did not realize this abstractness of his explanation in the Critique. However, Kant offered a much more enlightening arithmetic equation for the matter of mathematics syntheticity

“in a latter explaining his position to his disciple Johann Schultz, where he considers propositions like “3 + 5 = 2 × 4” (Correspondence, 10: 554–8). Here the failure of reciprocity is fully transparent, since it is obviously wrong to attribute the same content to the concepts, which involve different operations on different numbers, but the judgement is nevertheless true because the terms on each side “determine the same object” falling under them — that is, the magnitude to which they apply (Correspondence, 10: 555).”¹¹

Notes

1. Guyer, Paul, and R. Lanier Anderson. “Introduction to the Critique.” The Cambridge Companion to Kant’s Critique of Pure Reason, Cambridge University Press, New York, 2010, p. 77.
2. Ibid.
3. Ibid.
4. Ibid., p. 79.
5. Ibid.
6. Ibid., pp. 79–80.
7. Ibid., p. 83.
8. Ibid., p. 84.
9. Ibid.
10. Ibid., p. 90.
11. Ibid.