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The meaning of the term dialectic, as in Transcendental Dialectic, in the Critique Of Pure Reason, is obscure. This, mixed with the already complex text, makes this term difficult to assess.

A list of possible meanings that might fit:

  • Argumental: Reason would be an argumental mechanism, as Hegel's, producing assertions as the result of the process.

The second part of the transcendental logic [...] is called transcendental dialectic, not as an art of dogmatically arousing such il­lusion [...], but rather as a critique of the understanding and reason [...], in order to uncover the false illusion of their groundless pretensions and to reduce their claims to invention and amplification, putatively to be attained through tran­scendental principles, to the mere assessment and evaluation of the pure understanding[...]. (A64)

  • Discoursive: Reason would be the equivalent of a discourse, a structure of ideas, product of the tendency to make judgements that conduct to the unity of the self.

Only reason in its attempts to make out something about objects a priori and to extend cognition beyond the bounds of possible experience is wholly and entirely dialectical, and its illusory assertions do not fit into a canon of the sort that the analytic ought to contain. (B171)

...

Above we have called dialectic in general a logic of illusion. (B350)

  • Linguistic: When the Transcendental Dialectic process is associated with syllogisms.

Starting with the Transcendental Logic here, so, any help is appreciated.

UPDATE

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The genesis of the terms "analytic" and "dialectic" in Kant's time is discussed in Tonelli's Der historische Ursprung der kantischen Termini „Analytik“ und „Dialektik“, who concludes (my translation):

"The division of the Critique of Pure Reason into Transcendental Analytic and Transcendental Dialectic is undoubtedly after the process made by Darjes. Kant knew the Darjesian philosophy very well, he had dealt with it several times and stood in some details under the influence of Darjes and his school. The content of the Kantian transcendental analytics and dialectics is clearly quite different from that of the eponymous parts of the Aristotelian manuals; but the hypothetical and antithetical character the transcendental dialectic probably corresponds to the concept of logica probabilium or disputatrix. This analogy may have moved Kant to adopt the Aristotelian term."

Darjes is an 18th century author who published Introductio in Artem Inveniendi, seu Logicam theoretico-practicam, qua Analytica atque Dialectica... (1732). He defined the analytic as the "science of rules for discovering truths with certainty" and dialectic as the "science of discovering plausible truths". The "discovering" part separates him from contemporary Wolffians, who saw logic only as ars demonstrandi, and anticipates the understanding of Logic (and dialectic in particular) in later German idealism.

To the analytic Darjes assigned "concepts and definitions and their origin either a priori or a posteriori, judgments and sentences, intuitive judgments and the generality a posteriori, discursive judgments..." and to the dialectic "probability in general and as a way to dialectically find the truth, a way to find apparent terms or definitions, theoretical sentences a priori and a posteriori, philosophical hypotheses, hermeneutic probability, probable criticism, assessment of the likelihood of other people's opinions".

Kant has two types of analytic and dialectic, formal and transcendental. In both cases the division is between correct and incorrect employment of understanding and reason in their formal (general, syntactic) and transcendental (synthetic, semantic) functions. On Kant's "transcendental dialectic" there is more or less an agreement, see e.g. IEP, Kant’s Dialectic, Loparic, Kant's Dialectic, Guyer, The Unity of Reason and Rotenstreich, Kant's Dialectic.

It is the antithetical "logic of illusion" ("a sophistical art of giving to ignorance, and indeed to intentional sophistries, the appearance of the truth, by the device of imitating the methodical thoroughness which logic prescribes, and of using its "topic" to conceal the emptiness", A61), but of a special kind. It has to do with the tendency of pure reason to employ concepts of the understanding to unify knowledge beyond the boundaries of experience, and thereby fall into antinomies and "transcendental illusion". The reason preoccupies itself with directing understanding "towards absolute totality in the synthesis of conditions", which can never be guaranteed to terminate, but then transgresses into positing such absolute totalities as transcendent objects, as if this synthesis of appearances has already terminated. That's the transcendental dialectic. Here is Loparic:

"Yet, if we employ the principles of the understanding not as a canon for evaluating the empirical truth of synthetic propositions but as an organon, and consequently "venture, with the pure understanding alone, to judge synthetically, to affirm, and to decide regarding objects in general", our employment of the pure understanding becomes dialectical (B 88). It cannot be otherwise, because, as we have just seen, the a priori principles of the understanding provide universal conditions of synthesis of sensible objects and as such cannot offer the necessary basis for "passing judgments upon objects without distinction - upon objects which are not given to us, nay, perhaps cannot in any way be given" (B 88). An analogous dialectical illusion is generated through attempts to decide synthetical propositions regarding objects in general by employing solely formal logic and the transcendental principles of pure speculative reason (B 353)".

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  • Darjes is not known as a philosopher, apparently more - but not much more - as a theologian and a tad more so, as an economist. But as one prominent site said, his "writings were mostly unoriginal" even there. Oct 6 at 14:21
  • You say "Darjes assigned ... to the dialectic 'probability in general ...'; whilst Sgarbo in Kant and Aristotle writes "Hence from Djarjes did not take his conception of logic as logica probabilium as Tonelli suggested but the zetetic method of dialectic as cartharsis. Moreover, he states that "Kant did not employ the term analytic to signify a part of logic ... but in opposition to dialectic and within the contect of elaborating the doctrine of the Categories". This of course contradicts what you say that Darjes states: that it is a part of logic. Finally, I would add that ... Oct 7 at 1:44
  • All, in all, Darjes is a poor and misleading guide to Kant. Oct 7 at 1:46
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Dialectic as an method of verifying assumed truth

Dialectic is an old method (ancient Greeks) for establishing truth. Two people with different opinions in certain matter would engage in discourse, presenting their arguments and counter-arguments in order to persuade each other and the public. Latter it was carried over into literature, with Plato (for example) writing fictional dialogues (Socrates as main protagonist) in order to prove his point.

From all of that, dialectic got secondary meaning of any method that would verify and check long held beliefs and truth, possibly even overturning them. Kant uses this secondary meaning as his Transcendental Dialectic seeks to "uncover the false illusion of their groundless pretensions and to reduce their claims to invention and amplification" . In other words, Kant is trying to show limits of a priori knowledge (knowledge acquired without experience) and categories of the mind that also exist a priori (i.e. without prior knowledge) .

Kant (in his usual clumsy language :) ), argues that transcendental (metaphysical) knowledge practically does not exist (it is very limited), i.e. that understanding and reason could not invent anything new, and could not amplify already existing knowledge without experience. Term "transcendental" in Kant's work detonates something that exist outside of us and our experience, and roughly correlates with metaphysics.

What Kant allows in realm of transcendental is only debate about pure understanding, which corresponds with his aforementioned categories of the mind and a priori knowledge. And if we debate about objects a priori (without experience), such debate is only dialectical, i.e. serves only to verify (or denounce) existing beliefs, but could not yield anything new.

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    Dialectic can't be "any method that would verify [...] truth", because dialectic is itself a method.
    – RodolfoAP
    Oct 4 at 16:29
  • @RodolfoAP As I said in my answer, meaning of the word "dialectic" moved from original Greek meaning, describing dialogue as a method for verifying truth. From 18th century it became philosophical discipline that would seek for contradictions in proposed theory or established "truth" in order to verify or repudiate them. Therefore, any valid method for verifying truth would be in its essence dialectical.
    – rs.29
    Oct 4 at 18:15
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Dialectic in it's simplest incarnation means simply an argument or debate. This can be seen from the word it is derived from - dialektike - which means the art (techne) of (dia) speech (legein).

Philosophically, though, it's an argument towards truth and not an argument where either one or both protagonists are merely arguing to win - that would be the bad form of the art of rhetoric.

Now, it is important to understand the argument rather than stumble over the words by which the author titles a work. After all, a painting might be called 'a symphony in blue' and which might not even have any blue in it but is about an evocation of melancholy. This is why the blues singers of the US sing the blues. Here, it is more important to understand and appreciate the music rather than the term 'blues'. Likewise the relationship between the argument explained and the term 'argument' or dialectic. It's also important to set the argument in context - context is always important.

The SEP article has an excellant overview of the Transcendental Dialectic and it's worth quoting from them:

How are synthetic apriori propositions possible? This question is often understood to frame the investigations at issue in Kant's Critique of Pure Reason.

Here they announce that they are placing the entirety of the Critique in context and they suggest this question helps us do that.

In answer to this, Kant saw it fit to divide his response into three 1) How are the synthetic apriori propositions of mathematics possible? 2) How are the syntheyoc apriori propositions of science possible 3) How are the synthetic propositions of metaphysics possible?

Here we see Kant apply this question to three domains of knowledge: mathematics, science and metaphysics. This three domains have been normative since the time of Plato amd Aristotle and before them, Pythagoras.

In a systematic fashion, Kant responds to each of these questions. The answer to question one is found in the Transcendental Aesthetic and the doctrine of the ideality of space amd time. The answer to question three is the Transcendental Analytic where Kant seeks to demonstrate the essential role played by the categories in grounding the posdibility of knowledge and experience.

Here, he addresses the first two questions which, though important, is not kmportant to this discussion.

The answer to question three is the Transcendental Dialectic and is a resoundingly blunt conclusion: the synthetic apriori propositions that characterose metaphysics are not really possible at all.

So although in Kants estimation he found successful answers to the first two questions - and which to Aristotle would have counted as metaphysics - he was unsuccessful in the third question. Here, metaphysics means something very different to that of Aristotle, although related to it, and is closer to theology. Thus

Metaphysics, that is, is inherently dialectical.

That is, there is no end point to theology when argued rationally. This was already stated in Plato. Kant, in a way, is refining Plato's conclusion. This motivates Pascal's leap of faith and the sufi's where they find "a way and a means of striking a root through 'the narrow gate' in the depth of the soul out into the domain of the pure and unimprisonable Spirit which itself opens out into the Divinity". It's important to recall here that thought, and especially rational thought, is not the only faculty of the mind.

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    Didn't downvoted this, but it seems to me that the SEP article, which I've already checked and found incoherent, applies the term dialectic as begging the question: metaphysical pure knowledge would not be possible because some arbitrary metaphysical (dialectic, as argumental) judgements are not possible. And such misunderstanding is possibly due to the improper comprehension of dialectic.
    – RodolfoAP
    Oct 6 at 3:02
  • @RodolphoAP: It is no such thing. As an encyclopedia article it is written to be comprehensible to a large public, even if not philosophically knowledgeable in speacialist areas. The misunderstanding is on you side and has nothing to do with understanding the word dialectic. Why did Kant write this work? Merely to explain what 'dialectic' means? It's a critique. Oct 6 at 14:16
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My own answer was added, because this is clear and coherent now, following Conifold and Mozibur Ullah's guide (thanks!) and some more immersion in the Transcendental Logic.

In order to understand the sense of the term, it must be approached in the context of the whole kantian Doctrine Of The Elements.

Given that metaphysical knowledge had a bad reputation at the time, Kant's key concern is if metaphysics is:

  1. if it is possible at all, and,
  2. if possible, approach it a science.

So, he identifies and circumscribes with precision the sources of metaphysical and physical knowledge.

The first, metaphysical knowledge, is essentially identified as having synthetic a priori sources; that means, it is purely metaphysical. The second, physical knowledge, has a posteriori sources.

So, in order to systematize the analysis, Kant divides metaphysical knowledge in three parts:

  1. transcendentally aesthetic (as given by perception) knowledge (maths), which, Eureka! it is already considered a science;
  2. transcendentally analytical (as manifolded by understanding) knowledge (natural philosophy) which at the time is also considered scientific, and
  3. transcendentally dialectical (as illusory conclusions raised by reason) knowledge, which rests to be focused.

For that, it must be considered that "dialectical" at the time meant "that can't be proven" [1], or "that is probable but not certain" [2]. The word itself, "dialectical" (in Hegel's sense, a repeated sequence of thethical, antithetical and synthetical arguments), implies an iterative approach of propositions and conclusions, which Kant considers not to have limits, until reaching absurd due to improper interpretation and judgment of empirical facts, taking the form of illusions. Kant spent a lot of effort dissecting the source of rational illusions, finding solid arguments. The antinomies are the most precise proof of the consequences of the dialectic mechanism, which essentially lead to pure logical contradictions, exhibiting the dialectical thesis+synthesis=contradiction structure.

As a result, Kant concludes that transcendentally dialectical knowledge is impossible, do not exist as such, or, if it exists, it is flawed; essentially due to such dialectical nature.

In my own interpretation, this seems circular, given that Kant divides the metaphysical problem from an approach (canon, organon) which precisely allows one of such parts (the organon, where God or the infinity of the universe seem to fit logically) to be possible and the other as impossible. In simpler words, the Transcendental Dialectic targets what is not science, and it is only probable, in order to say that it cannot be metaphysically (or transcendentally) proven.

[1] Paul Rabe: https://www.researchgate.net/publication/268814181_Kant_Rabe_e_la_logica_aristotelica

[2] Joachim Georg Darjes: logica probabilium or disputatrix, as per Conifold's answer, https://www.jstor.org/stable/24356826

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