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First, some historical context:

The term apperception originates with René Descartes in the form of the word apercevoir in his book Traité des passions. Leibniz introduced the concept of apperception into the more technical philosophical tradition, in his work Principes de la nature fondés en raison et de la grâce; although he used the word practically in the sense of the modern attention, by which an object is apprehended as "not-self" and yet in relation to the self.

Immanuel Kant distinguished transcendental apperception from empirical apperception.

The wikipedia entry is confused on the rest, but on the separate entry on transcendental apperception

  1. All experience is the succession of a variety of contents (an idea taken from David Hume).

  2. To be experienced at all, the successive data must be combined or held together in a unity for consciousness.

  3. Unity of experience therefore implies a unity of self.

  4. The unity of self is as much an object of experience as anything is. Therefore experience both of the self and its objects rests on acts of synthesis that, because they are the conditions of any experience, are not themselves experienced.

  5. These prior syntheses are made possible by the categories. Categories allow us to synthesize the self and the objects.

First, is this a good summary of what Kant means by transcendental appercetion? Secondly what is empirical apperception and what was Kants rationale for introducing the distinction?

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When the empirical apperception functions, it allows you to conceptually recognize this or that particular object, as a unified object. You sense a manifold of stuff, but are able to combine what you sense into a object (say a unique, specific cup of coffee). Kant argues that this presupposes the transcendental apperception, that in itself can have no objects, but conditions and apply rules, that allow you to recognize an object as a an object.

This entry from G. J. Mattey's Kant Lexicon seems to make a clear differentiation. I highlight parts of it:

The notion of apperception is introduced in the Transcendental Deduction of the categories of the understanding and plays a central role therein.

(...)

Generically, apperception for Kant is self-consciousness. It may be either empirical or pure. In the first-edition (A) Deduction (A107), Kant equates empirical self-consciousness with inner perception or inner sense, whose object is succession of states of ourselves in time. “This consciousness of oneself is merely empirical and always mutable; it can give us no constant or enduring self in the flow of inner appearances.”

By contrast, pure apperception does not have anything experienced as its object. Moreover, unlike empirical intuition it is productive. In the second-edition (B) Deduction, Kant claims that it “produces the presentation I think” (B132). It is also “immutable” (in contrast to empirical apperception as described at A107). It is “numerically identical” (A132) or “one and the same in all consciousness” (B132). A further feature is that “it cannot be accompanied by any further presentation” (B132). Because of these features, pure apperception is also called “original” apperception.

One crucial feature of pure apperception is its unity or thoroughgoing identity. The unity of pure apperception is described as “transcendental” because it is an a priori condition for the presentation of objects. The second key feature of pure apperception is its necessity. The unity of apperception can be either analytic or synthetic.

According to this very detailed entry in the Stanford Encyclopedia of Philosophy, the crucial passages related apperception is "blindingly difficult". In the Stanford article it is also disputed, that we can equate apperception with self-consciousness.

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The transcendental apperception is the "I think" that goes with all my representations. That is, when I perceive a table, I know that representation belong to one "I". On the other side, the empirical apperception is to recognise the identity between the phenomenon of the table and my representation of the table.

Kind regards.

EDIT: and yes, I think that that description is pretty good.

EDIT 2: Aputsiaq correctly noted one mistake at my answer: "the empirical apperception has to do with the external sense". That´s totally incorrect as for Kant:

Unsere Vorstellungen mögen entspringen, woher sie wollen, ob sie durch den Einfluß äusserer Dinge, oder durch innere Ursachen gewirkt seyn, sie mögen a priori, oder empirisch als Erscheinungen entstanden seyn; so gehören [99] sie doch als Modificationen des Gemüths zum innern Sinn, und als solche sind alle unsere Erkentnisse zulezt doch der formalen Bedingung des innern Sinnes, nemlich der Zeit unterworfen, als in welcher sie insgesamt geordnet, verknüpft und in Verhältnisse gebracht werden müssen. (KrV, A 98-99)

The influence of external things (den Einfluß äusserer Dinge) affects the inner sense, as a modification of time. As Allison (Kant´s Trascendental Idealism, 2004) notes, the distinction between the two "apperceptions" is troublesome as Kant sometimes tends to identify inner sense with empirical apperception. Anyway, intern sense has nothing to do with apperception.

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    The inner-external distinction seems invalid according to sect. 4.1 in plato.stanford.edu/entries/kant-mind, "There are two kinds of consciousness of self: consciousness of oneself and one's psychological states in inner sense and consciousness of oneself and one's states via performing acts of apperception." – Aputsiak Aug 27 '13 at 21:31
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    you are right. I wanted to write something else. I´m going to correct it. – Strabo Aug 28 '13 at 2:25

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