The Critique of Pure Reason reading notes

Categorization of knowledge

But although all our knowledge begins with experience, it does not follow that it arises from experience.

Kant categorized knowledge as follows:

empirical knowledge has its sources a posteriori, that is, in experience.
a priori knowledge is “independent of experience and even of all impressions of the senses”.
a priori knowledge is pure if nothing empirical is mixed with it. That is, by pure a priori knowledge, we refer to knowledge absolutely independent of all experience, not knowledge independent of this or that experience.

Analytic vs synthetic judgements:

A judgement is analytic (elucidatory) if the predicate of the subject is contained in the subject. e.g. “All bachelors are unmarried” is an analytic judgement since the predicate “unmarried” is contained in the subject “bachelor”.
A judgement is synthetic (expansive) if the predicate is not contained in the subject. e.g. “All bodies are heavy” is a synthetic judgement since the predicate “heavy” is not contained in the concept of a “body”.
From mere concepts we can only obtain analytic but never any synthetic knowledge (B64).
Empirical judgements are all synthetic.
All (pure) mathematical judgements are synthetic.

The core question of the entire Critique is: How are synthetic judgements a priori possible?

Transcendental Aesthetic

Transcendental Aesthetic is the study (science) of the a priori condition (principles) of sensibility.

Some definitions:

Whatever the process and means may be by which knowledge refers to its objects, intuition is that through which it refers to them immediately.
The capacity (receptivity) to obtain representations through the way in which we are affected by objects is called sensibility. From B61, sensibility is the receptivity of our capacity of knowledge. Even if we could see through, and to the very bottom of, that appearance, this receptivity would remain for ever altogether different from knowledge of the object in itself. We are so constituted that our intuition can never be other than sensible (B75).
The effect produced by an object upon the capacity for representation, insofar as we are affected by the object, is sensation.

Some properties of the above concepts:

Objects are given to us by means of our sensibility.
Sensibility supplies us with intuitions.
An intuition that refers to an object through sensation is called empirical.
The undetermined object of an empirical intuition is called appearance.

There is a crucial distinction between empirical intuition, which is related to sensation, and pure intuition, which is not and is independent from experience. Appearances are constituted by two elements:

Matter, that in an appearance which corresponds to sensation.
Form, that which brings about the fact that the manifold of the appearance can be ordered in certain relations.

Space and Time

Time cannot be intuited externally, any more than space can be intuited as something within us. What then, are space and time? Are they things which actually exist? Are they only determinations or relations of things, but such as would belong to things even if they were not intuited? Or are they determinations and relations which adhere only to the form of intuition and therefore to the subjective nature of our mind, without which these predicates of space and time can never be ascribed to anything at all?

The aesthetic is dedicated to space and time. Kant makes a distinction between metaphysical exposition and transcendental exposition:

Exposition refers to the distinct, though not comprehensive representation of what belongs to a concept.
An exposition is metaphysical if it contains that which exhibits the concept as given a priori.
A transcendental exposition is the explanation of a concept as a principle from which the possibility of other synthetic a priori knowledge can be understood.

The two metaphysical expositions of space and time are similar. They both state that space and time:

are not empirical concepts derived from outer experiences.
- Neither simultaneity nor succession would enter our perception if the representation of time did not underlie them a priori.
are necessary a priori representations which underlie all outer intuitions.
- It is impossible to have a representation of there being no space, though one can very well think of space without objects to fill it.
- Similarly, we cannot remove time itself from appearances in general, though we can quite well take away appearances from time.
are not discursive, or a general concept of the relations of things in general, but pure intuition.
are represented as infinite magnitudes.

The transcendental expositions state:

Space grounds the possibility of geometry
Time grounds the possibility of the concept of alteration and motion.

In addition, time is an a priori form of inner intuition, while space is an a priori form of outer intuition. Time is the formal a priori condition of all appearances in general. Space, as the pure form of all outer intuition, is restricted, as an a priori condition, to just outer appearances. But as all representations, whether they have as their objects outer things or not, themselves belong, as determinations of the mind, to our inner state.

Time is nothing but the form of inner sense, that is, of our intuition of ourselves and of our inner state.
Take away from time the special condition of our sensibility, and the concept of time vanishes as well; time does not adhere to the objects, but only to the subject that intuits them.

The transcendental aesthetic cannot contain anything more than space and time. All other concepts belonging to sensibility presupposes something empirical. For example, the concept of alteration requires the perception of something that exists and the succession of its determinations, which can only be found in experience. Motion, also, for example, presupposes something movable, but there is nothing movable in space itself, so it must be something found in space only through experience.

Further notes

We can only know an object’s appearance, not what the object is in itself.

What the objects may be in themselves would never become known to us even through the most enlightened knowledge of that which alone is given to us, namely, their appearance.

We commonly distinguish between appearances that holds for every human being from appearances that are more subjective (that which belongs to their intuition only contingently, being not valid for the sensibility in general), thinking the former is the “truth” (that is, representing the object in itself) and the latter its “appearance”, but Kant believes this distinction to be merely empirical, and both of these are merely appearances (see B63 rainbow example). This is because space and time are merely subjective conditions of all our intuition, and that in relation to these conditions all objects are therefore mere appearances. Therefore, nothing can ever be said about the thing in itself which may underlie these appearances. Note: by this, he is not saying that these objects are a mere illusion.

Transcendental Logic

We must have a way to seek out pure concepts systematically so that we know they are all covered instead of just seeking them out as we think of it. Kant expresses this by saying that understanding is “an absolute unity”. He says that pure concepts must originate from a source, and “must be connected with each other according to one concept or idea”. This is an unproved assertion. But he emphasizes this in a few places:

But such a connection supplies a rule by which we are enabled to assign its proper place to each pure concept of the understanding and by which we can determine in an a priori manner their systematic completeness. Otherwise we should be dependent in these matters on our own discretionary judgment or merely on chance.
The functions of the understanding [i.e. the a priori concepts of understanding] can be discovered in their completeness, if it is possible to state exhaustively the functions of unity [i.e.the forms of relation] in judgments.

Our knowledge springs from two fundamental sources of our mind (both must be present for knowledge to arise):

to receive representations (receptivity of impressions), through this an object is given to us, this is sensibility.
the faculty of knowing an object through these representations (spontaneity of concepts), through which an object is thought, this is the understanding.

Thoughts without content are empty, intuition without concepts are blind. (B75)

Pure intuition vs pure concept (see Categorization of knowledge chapter for definition of pure vs empirical):

Pure intuition (contains only the form under which something is intuited,
Pure concept contains only the form of thinking an object in general.

Kant’s articulation of logic divides into two main parts:

general logic (also called elemental logic): contains the absolute necessary rules of thought without which the understanding cannot be used at all
logic of the particular use of the understanding (also can be called organon of a science): contains rules for how to think correctly about a certain kind of objects

General logic divides into two parts:

pure: remove all empirical conditions, only deals with a priori principles.
applied: directed to the rules of the use of understanding under subjective empirical conditions, it is simply a cathartic of the common understanding

Kant introduces another form of logic: transcendental logic. What is transcendental logic? It involves content, but only the a priori condition of it.

Important remark on transcendental logic: “Not every kind of a priori knowledge is transcendental; only that kind by which we know that and how certain representations (intuitions or concepts) can be used or are possible completely a priori should be called transcendental , that is, the a priori possibility or the a priori use of knowledge.” (B81)

About truth: Truth is the agreement of knowledge with its object, which means it must necessarily involve an object, so it is pointless to look for an indicator for truth that is independent of the object. Even if a piece of knowledge agrees with our rules of understanding (it does not contradict itself), it can still contradict the object, so logic can go no further than this. This reminds me of people on Quora who ask whether the value of pi is true or similar, it seems this paragraph is an apt response.

What is judgement?

Judgment is … the mediate cognition of an object, hence the representation of a representation of it.

(1) QUANTITY OF JUDGEMENTS: Universal, Particular, Singular

(2) QUALITY: Affirmative, Negative, Infinite

(3) RELATION: Categorical, Hypothetical, Disjunctive

All the relations of thought in judgements:

(a) CATEGORICAL: relation of the predicate to the subject (only two concepts considered)

(b) HYPOTHETICAL: relation of the ground to its consequence (e.g. if there is perfect justice, then the obstinately wicked is punished. Whether these two propositions are true in themselves is irrelevant; it is only the consequence which is thought by this judgement.)

(c) DISJUNCTIVE: relation of the subdivided knowledge ad the collected members of the subdivision to one another (contains the relation of two or more propositions to each other, a relation not of consequence, but of a logical opposition, insofar as the sphere of the one proposition excludes the sphere of the other, and yet at the same time of community, insofar as the two propositions together fill the whole sphere of knowledge proper. e.g. We can say: The world exists either through blind chance, or through inner necessity, or through an outer cause. Each o these propositions occupies a part of the sphere of all possible knowledge concerning the existence of a world in general, while all of them together occupy the whole sphere, i.e. the options are complementary and add up to a probability p = 1.)

(4) MODALITY: Problematic, Assertoric, Apodictic

Contributes nothing to the content of the judgement, but concerns only the value o the copula in relation to thought in general.

(a) PROBLEMATIC judgements: affirmation or negation is taken as merely possible (optional), expresses only logical (not objective) possibility, that is, it expresses a free choice of admitting such a proposition, and a purely optional admission of it into the understanding. e.g. in a hypothetical syllogism the antecedent is problematic in the major premise, but asertoric in the minor.

(b) ASSERTORIC judgements: affirmation or negation is taken as actual (true), deals with logical actuality or truth (merely asserts that something is (or is not) the case, in contrast to problematic propositions which assert the possibility of something being true)

(c) APODICTIC judgements: affirmation or negation is taken as necessary, expresses logical necessity (assert things which are necessarily or self-evidently true or false)

e.g. from wikipedia: “Chicago is larger than Omaha” is assertoric. “A corporation could be wealthier than a country” is problematic. “Two plus two equals four” is apodeictic.

[…] synthesis is what gathers the elements for knowledge and unifies them into a certain content. It is to synthesis, therefore, that we have to direct our attention first if we want to form a judgement about the first origin of our knowledge.

e.g. on the concept of cause, which implies a special kind of synthesis: we can’t say it is a priori clear why appearances should contain something of this kind. We also cannot just conclude from appearances: “it must either be founded completely a priori in the understanding, or must be given up altogether as a mere hallucination”. This is because this concept requires that an event follows another necessarily and according to an absolute rule. Appearances supply us with cases, where such a thing usually happens, but never a rule according to which the result is necessary. The strict universality of a rule can never be a property of empirical rules.

There are two conditions under which alone knowledge of an object is possible:

intuition, through which the object is given, though only as appearance.
concept, through which an object is thought that corresponds to this intuition

An a priori concept that did not refer to experience would only be the logical form of a concept, but not the concept itself through which something is thought. If therefore, there exist any pure a priori concepts, they cannot indeed contain anything empirical; they must, nevertheless, all be a priori conditions of a possible experience, for on this ground alone can their objective reality rest.

Of the Deduction of the Pure Concepts of the Understanding, Of the A Priori Grounds for the Possibility of Experience

[…] keep in mind that appearances are not things in themselves, but the mere play of our representations, which in the end are determinations of inner sense.