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I need help in understanding Kant’s Critique of Pure Reason § 18 and 19. What Kant does there suggests an alternative to the idea that a concept represents a different object. So the question is about the relation between conceptions and objects and how they are related according to Kant in the end. I’d be really thankful for an answer or a clue, any keywords that will lead me forward. I mean I’ve read these passages many times now, it’s just that I fail to get his main point. Also, I’m reading it in German since I’m a native German speaker, but ironically I think reading Kant in English might be easier.

EDIT: I think it’s always useful to remind ourselves of the basis of Kant’s ideas. I’m sorry I wasn’t that precise on the pages I’m referring to, I’m referring to the chapter about consciousness and objectivity: B131-B140+A104-A106 and A108-A110 (I’m not sure if this might help since I just have my professors PDFs, and not the actual book). Anyway, he there again questions the relation between object and concept but this time he refers to the objective unity of the self and to the transcendental deduction. So what I’m asking myself is in what relational role they play between a concept and their object.

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Caveat

I'm no expert in the Critique of Pure Reason, and less so in [Kritik der reinen Vernunft], but maybe I can nudge you a bit, hopefully not in the wrong direction.

Long Answer

It's fair to say that Kant does not advocate direct realism. In fact, transcendental idealism might be considered the culmination of thousands of years philosophical debate and the cornerstone of all current Western philosophy, a thesis put forth by Simon Critchley in his Continental philosophy: a very short introduction. What's important is the overarching theme. From WP:

In the "Transcendental Aesthetic" section of the Critique of Pure Reason, Kant outlines how space and time are pure forms of human intuition contributed by our own faculty of sensibility. Space and time do not have an existence "outside" of us, but are the "subjective" forms of our sensibility and hence the necessary a priori conditions under which the objects we encounter in our experience can appear to us at all. Kant describes time and space as "empirically real" but transcendentally ideal.

Kant argues that the conscious subject cognizes the objects of experience not as they are in themselves, but only the way they appear to us under the conditions of our sensibility. Thus Kant's doctrine restricts the scope of our cognition to appearances given to our sensibility and denies that we can possess cognition of things as they are in themselves, i.e. things as they are independently [sic] of how we experience them through our cognitive faculties.[citation needed]

This property of physical objects can be captured in the short phrase Ding an sich. From WP:

In his doctrine of transcendental idealism, Kant argued the sum of all objects, the empirical world, is a complex of appearances whose existence and connection occur only in our representations.2

I'm going to warn you that I'm interpreting the English interpretation of a German work written in the context of philosophical discourse that happened almost 250 years ago, so I'll do my best to keep to his theory, but may "translate" a bit into modern philosophical terminology for perspective. As far as I'm concerned, indeterminacy of translation is a lot more pervasive than mere scientific theory and philosophy across time is an hermeneutical practice. I'll make heavy use of use-mention distinction to try to keep things straight. So, let's sum up §18&19 which defines the 'Objective Unity of Self-Consciousness' and describes the commonality of the 'Logical Form of All Judgements'.

§18

The 'objective unity of self-consciousness' is distinct from 'subjective unity of consciousness' the latter is derived from the 'sensibility of inner sense' which combined with the 'manifold of intuition' is 'empirical'. The 'empirical unity of consciousness' creates contingent truths through the 'association of representations of objects. The 'original unity of consciousness' alone is 'objective', provides 'intuition in time', relates to 'I think', and to 'pure synthesis of understanding' providing a priori truth for 'empirical synthesis' which has 'subjective validity' only.

§19

Logicians claim 'judgement' is a 'representation of a relation between two concepts'. 'Judgement' is integrating knowledge into the 'objective unity of apperception'. The copula 'is' expresses contingent truths such as 'Bodies are heavy' which is a 'synthesis of intuitions' and not an 'empirical intuition' necessarily. Such contingent truths are 'judgements' which are 'objective relations' and different from other 'representations' which are 'subjective relations'. 'If I carry a body, I feel the pressure of its weight' is subjective. 'It, the body, is heavy' is objective. The 'object' is a combination of these claims and are not both 'perceptions'.

Huh?

What we have going on here is Kant beginning to wrestle with intersubjectivity in order to deal with the dichotomy of the subjective and the objective. First, let's go back to §10 where Kant essentially borrows Aristotelian categories. The basic claim is "general logic... abstracts from all content of our knowledge, but expects that representations will come from elsewhere" to build concepts of analytic truth. This is the analytic-synthetic distinction. Analytic truths are word-based truths, if you will, and synthetic truths come from experience of the world. The way I understand this, is that general logic deals with the logical abstractions of language what today is called the extension of language, where as the synthetic truths are related to what we now call the intensions of the words. So, transcendental logic is that that goes beyond mere general logic, it is a logic that seeks to look at both the extensional and intentional aspects of claims.

But claims about the world aren't "direct". They are filtered through the objective Categories of the mind which are categories that are built by the mind universally. Thus, when we talk about an object existing in space and time, we are referring to Quantity, Quality, Relations, and Modalities all of which are properties of the mind. We can't know an object directly, but rather in terms of these categories which all minds begin with. This is the elementary and objective set of concepts our minds intuit with. In more modern language, objects in time-space to Kant was part of our collective naive physics.

Now, the 'transcendental unity of self-consciousness is the Cartesian consciousness that is connected in cogito, or if you prefer Ich denke. Aber Ich denke has to be seen as a product of analytic and synthetic truths, as well as a priori and a posteriori truths according to Kant. What is objective to Kant is knowledge derived from the synthetic a priori because it is both objective and before subjective experience which is built from intuitions and inner sensibility. But how is that a person comes to have both objective and subjective truths as an integrated person? This is the role of his transcendent logic, which takes the subjective truths born of objective Categories and raises them to the level of Ich denke, the self-aware consciousness, out from the 'manifold of intuition' using 'inner sensibility'. So, a transcendent understanding of an external object is a combination of both its objective-pre-intuitional a posteriori representation and its relationship to other a priori truths such as synthetic a priori facts about quantity or quality. And putting together all of these different classes of truths is the broad notion of 'judgement' which consists of the 'objective unity of apperception of the concepts' contained within. Roughly speaking, this is a theory of intersubjectivity that can be seen in both "analytical" and "Continental" philosophies today. He then goes on in §20 to explain claim why intuition relates to the categories and is derived from them.

Es Tut Mir Leid?

I don't know if that helps at all. The way I see it, what Kant is attempting to do is differentiate how we know both rationally and empirically and explain how we integrate that knowledge, when the rational seems to come from extensional language containing abstract logical relations at the same time explaining how we can all experience objects differently and subjectively both through conscious and intuitional thought. He's trying to dissolve the objective-subjective dichotomy resulting from the dualism of mind by creating a model that grounds the objective in bodily truths (categories), explains how knowledge can be spontaneous (manifold of intuition), demonstrate that some relations seem universal (a priori), and others highly particular (a posteriori). Also, some truths are very much based in language (analytic) and others are very much based in our knowledge of the world (synthetic). All in all, his terminology is still quite popular in philosophy.

I hope this helps, and if I seem more confused than you, it might very well be that I am. Viel Glueck!

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One of Kants objectives is naturalising philosophy and this means Plato's philosophy. Whereas Plato's ideas were in some intelligible realm, Kant actually places them in the human mind and has them be the basis of how we understand the world: the basic categories of time and space. And categories built on top of this: measure, possibility, impossibility and causality etc.

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