Preface to the First Edition

Reason functions by beginning with fundamental principles that allow it to make sense of nature. As it progresses, it abstracts from these principles and reaches areas that are more and more remote; this continues until reason reaches an inevitable stage at which it asks itself questions that it cannot answer. This stage is known as metaphysics.

Metaphysics was once seen as the 'queen' of all the sciences; it is now despised and seen as an outcast.

When metaphysics still reigned, dogmatists allowed it to 'rule despotically'; but the dogmas of the different dogmatists clashed to such an extent that skeptics came along and pushed all metaphysics and 'civil unity' into disarray. Following this, the dogmatists would come again and try to piece things back together, in a multitude of ways.

Some philosophers, such as Locke, tried to test the 'lawfulness' of these different dogmatisms, by examining through a 'physiology' of human understanding. But even though this rendered dogmatic thought as suspicious, dogmatisms continued to assert themselves, since 'this genealogy was attributed to [metaphysics] falsely'.

Following these pursuits, all of which were in vain, the general attitude towards metaphysics was one best described as indifferentism. But it is a kind of indifferentism that leads to metaphysical confusion in the first place! The indifferentists, therefore, always fall back into metaphysics. This phenomenon ought to be examined.

In this examination, a kind of 'court of justice' will be established, for reason to arrive at a state of self-knowledge, wherein it can 'secure its rightful claims while dismissing all its groundless pretensions ... according to its own eternal and unchangeable laws'. This court is the critique of pure reason, the critique of the faculty of reason insofar as it can arrive at cognitions independently of experience. It is therefore, at the same time, an answer to the question of whether or not metaphysics is possible, and of what the origin of metaphysical thinking is.

No questions in the critique of pure reason will be avoided by using reason's inadequacy as an excuse; instead, they will be reformulated so as to accommodate the nature of reason.

'Comprehensiveness' is the 'chief aim' of the critique, and as such it answers all metaphysical problems. In fact, pure reason is rightly treated as a 'pure unity'; as such, if any aspect of it was unsuitable for one question, it would be discarded for all other questions: if it fails for one question, it is unreliable for any others.

Kant acknowledges that his promises sound immodest; however, he argues that they are more modest than the philosophers that claim to possess knowledge on the human 'soul' or on the beginning of the universe. Kant merely promises to gain an exhaustive understanding of the human faculty of reason. The critique of pure reason only concerns itself with the claims that can be made with necessity.

The book is difficult, but Kant suggests that 'many a book would have been much clearer if it had not been made quite so clear': giving too many examples, illustrations, and reductions of complicated topics into common concepts might help to clarify a part, but will only confuse the whole.

Preface to the Second Edition

Is the examination of the cognitions of pure reason a science? If, while we are carrying it out, we find ourselves having to scrap our progress and go down another path; if we get completely stuck after making sensible preliminaries; if we can never agree on how to establish our path in the first place - in any of these cases, we will realise that it is not a science.

Logic, for example, has never had to reroute itself since the time of Aristotle. Nor has it found the need to take extra steps - apart from those which have been psychological, metaphysical, or anthropological, all of which Kant dismisses as distortions of the boundaries of logic as a science. Logic has had such success because it concerns itself only with the form of understanding, and is abstracted away from all potential objects of cognition.

On the other hand, it is more difficult when a science considers not only cognition but the objects of cognition. Whenever a science has an object, it must involve some a priori cognition: either the object is determined and its concept is brought about, or the object is actualised. The former kind of reasoning is theoretical; the latter is practical. In both kinds of reasoning, the kind of object that the reasoning concerns itself with is determined a priori, or else the reasoning would end up being misapplied, 'for it is bad economy to spend blindly'.

Unlike mathematics and the natural sciences, which have managed to earn the status of being scientific, metaphysics has always lost itself - despite the aforementioned fact that metaphysical thinking seems to be forced upon us by nature. We might, therefore, reflect on what kinds of thinking have been beneficial for mathematics and the natural sciences, and approach metaphysics with a similar kind of thinking.

We have typically assumed that all cognition must conform to its objects; however, at least in the case of metaphysics, this has got us nowhere. At least for the sake of experiment, then, it is worthwhile to approach metaphysics with the assumption that objects must conform to our cognition. This would allow us to have an a priori idea of what our objects are which, as previously argued, is required for scientific thinking.

Approaching metaphysics in this (scientific) manner will allow us to determine if we are getting an accurate picture of the objects of metaphysics, simply by asking: does this conform to the rules of our cognition? Admittedly, this strength also comes with the disadvantage that we can only make a priori cognitions of objects regarding those aspects that we ourselves have 'put into' them. This also means that we cannot go beyond the boundaries of possible experience which, at least traditionally, is the purpose of metaphysics.

If we go along with the assumption that objects conform to our knowledge, then we must accept that all of our cognitions only reach the appearances of things; there is therefore an unconditioned aspect of things in themselves which we cannot cognize, and it is this unconditioned that drives us towards the confusions of metaphysical thinking. Importantly, the notion of the unconditioned is only possible if we accept that objects conform to knowledge; if we make the opposite assumption, the notion of the unconditioned is contradictory.

Having accepted that theoretical reason cannot cognize the unconditioned, Kant claims that we must test whether or not practical reason can make such cognitions:

"What still remains for us is to try whether there are not data in reason's practical data for determining that transcendent rational concept of the unconditioned, in such a way as to reach beyond the boundaries of all possible experience, in accordance with the wishes of metaphysics, cognitions a priori that are possible, but only from a practical standpoint". (Bxxi)

This is done in the Critique of Practical Reason.

What will be the results of a critique of pure reason? Firstly, we will be given a system of principles in which each principle depends upon all the others; no principle can be taken with certainty on its own unless it works in relation to the entirety of pure reason. Since these principles are based solely off of the structure of human reasoning, they will never be appended to or subtracted from; as such, once (critical) metaphysics has established itself as a science, it can be laid 'down for posterity', as a complete framework. This is a unique luxury that the natural sciences don't have.

Admittedly, there is a negative side to critical metaphysics, in that it is strict about the limitation of pure reason, 'teaching us never to venture with speculative reasoning beyond the boundaries of experience'. However, it is actually positive insofar as it completely prevents the negative consequences of pre-critical metaphysics: namely, the tendency to go beyond the limits of experience and thereby limits (or even prevents) the use of practical reasoning. As such, the critique of pure reason is giving a greater respect to the requirements of practical moral reasoning.

It has already been established that the critique will show that we cannot cognize things as they are in themselves. This doesn't mean, however, that we cannot even think of things in themselves; we can, because without such an ability we would have to assume that there are representations without things to represent. This twofold conception of the object also allows us to consider particular things in twofold manners - for example, to consider the human soul as sensibly determined by the laws of nature, but intellectually (in itself) free. I cannot cognize myself as free, but I can think myself to be that way, without any contradiction. This is a requirement of moral practical reasoning, because without freedom there can be no moral decisions. It is also, Kant will argue, necessary to assume the existence of God and immortality for practical reasoning to succeed, and these assumptions cannot be made without the limitations that the critical project places on pure reason.

The critique of pure reason is therefore a beneficial project. Kant defends this point further by pointing out that the endeavours of pre-critical metaphysics - ontological and cosmological arguments for God, defences of free will on the basis of 'powerless distinctions', and so on - have never actually benefitted the public, only the scholars. As such, the systematic disproval of these endeavours is only a drawback to scholars who wish to speculate endlessly without concern for social or pedagogical benefit.

Critical philosophy isn't critical of dogmatic procedure - the procedure of pure reason is always dogmatic, since it secures its principles with necessity. Instead, it is critical of dogmatism, which can be defined as dogmatic procedure 'without an antecedent critique of its own capacity'. According to Kant, such an antecedent critique of the 'organ' of reason was the only thing preventing Wolff from completing a systematic and scientific metaphysics.

Introduction (revised in B)

I. On the difference between pure and empirical cognition.

All cognition begins with experience, since it is only objects that can stimulate the senses and thereby produce a representation in the mind. However, although cognition begins with experience, it doesn't necessarily arise from experience. This is because our cognitions may be composed of both that which is sensuously intuited, and also that which our cognitive faculty (the understanding) provides, after merely being "awoken" or "provoked" by sensuous intuition. With philosophical practice, we can separate cognitions into their sensory and rational components.

We have established that all cognitions are brought to stage with experience, and that they are built up of components that are either sensuously intuited, or provided by the cognitive faculty. We may still ask, then: are there any cognitions which are composed entirely of that which our cognitive faculty provides? Though these cognitions would begin with experience - that is to say, they will be "provoked" by experience - they may have their content determined entirely independently of that experience.

This question can be rephrased as, are there any a priori cognitions? Such cognitions are distinguished from a posteriori cognitions, which have their sources in experience.

But, in fact, this definition of a posteriori cognitions isn't precise enough, because some cognitions that have experience as their source seem to be arrived at a priori, because the experiential sources aren't immediate but instead derived from general rules that are "borrowed from experience". Here, Kant seems to be alluding to Hume's notion of "custom". Kant's example is that we know a priori that a house without foundations will collapse, without having to have seen this particular house have its foundations undermined. But this is not known purely a priori, because - as it has been explained - the 'rules' it relies upon are borrowed from experience.

Following this clarification, Kant categorises different cognitions as follows:

• Empirical or a posteriori cognitions are those that are only possible through experience. e.g. "This house is big".
• A priori cognitions are those that can be justified without referring to experience. e.g. "Every alteration has its cause" - note how "alteration" is an empirical concept, but the cognition itself is still a priori, because "alteration" doesn't refer to an empirical particular.
• Pure a priori cognitions are those in which nothing empirical is 'intermixed'. e.g. 2 + 2 = 4

Note: the distinction between "pure" and "impure" a priori cognitions seems to be very unclear in Kant. As such, the categorisation above might not be entirely accurate. Kant introduces the categorisation in B3.

II. We are in possession of certain a priori cognitions, and even the common understanding is never without them.

If a proposition is necessary, then it must be a priori, because experience tells only that a thing is how it is - it doesn't teach us that the thing cannot be otherwise. In addition to this, if a proposition has genuine necessity - that is, there cannot be any conceivable exception to it - then it must be a priori, since experience only gives us an 'assumed' universality through induction.

There are examples of statements that are entirely a priori and, therefore, (truly) universal and necessary. The most obvious examples are the statements of mathematics. "Every effect has its cause" is also an example because the understanding's concept of "cause" undeniably contains the concept of "effect"; if they were to be in any way hypothetically separated, as they are in Hume's work, and derived from mere "subjective necessity" - that is, custom or habit.

Even without giving examples, the existence of a priori cognitions cannot be doubted, because they are a necessary condition for all experience. What would give any empirical experience its certainty if all principles it relied upon were also empirical? The existence of a priori cognitions is also a necessary condition for the existence of concepts: take any empirical concept, such as a body, and remove everything empirical from it (the colour, texture, weight, etc) - the fact that it still exists in space cannot be removed. This is because space is an a priori intuition. As such, it is undeniable that our understanding possesses certain a priori cognitions.

III. Philosophy needs a science that determines the possibility, the principles, and the domain of all cognitions a priori.

As already established, the human understanding typically desires to find cognitions that cannot be verified by any possible experience and that therefore 'go beyond' the realm of experience. The study of these cognitions is metaphysics, and it is consistently undertaken as a form of dogmatism, without any preliminary critique of the organ (i.e. reason) that it uses. The topics it typically concerns itself with are God, immortality, and freedom.

Since this kind of dogmatism, by very nature of the cognitions it focuses on, cannot be refuted by experience, it is only a critique of pure reason that can actually refute it, unless a dogmatist herself realises that her work has reached an obvious contradiction. This is likely to happen since, when a dogmatist frees themselves of the bounds of experience, their 'drive for expansion' no longer has any limits. This drive is so strong that some people may speculate to an absurd extent, and then make excuses to justify the groundlessness of their speculations.

The critique of pure reason doesn't need to worry about this kind of trouble, because it is primarily concerned with analysing concepts that we already have of objects. This gives us illuminations and clarifications of our existing cognitions, which can themselves be treated as new cognitions for their ability to ground a future metaphysics. Real progress is made, therefore, by these a priori cognitions, and at first this seems contrary to the very definition of an a priori cognition. However, Kant points out, there are in fact some a priori cognitions which aren't simply tautological. The next section deals with the distinction of these cognitions from 'ordinary' a priori cognitions.

IV. On the difference between analytic and synthetic judgements.

In all judgements (combinations of intuitions and cognitions of the understanding) where the relation is given between a subject and a predicate of the subject, there are two possible relations. Firstly, predicate B inherently belongs to subject A. Secondly, predicate B lies outside of subject A. The former kind of judgement is called analytic and is a judgement of clarification; the latter is called synthetic and is a judgement of amplification.

Judgements of experience are always synthetic, since analytic statements needn't go beyond a mere clarification of a concept, and experience appends rather than clarifies. In other words, experience works to amplify cognition.

Synthetic judgements that are a priori, however, must be amplified differently. If I cannot resort to experience to amplify concept A in an a priori judgement, what do I use instead? As an example, take the proposition that "everything that happens has a cause" (note the different wording to the previous [analytic] proposition, "every alteration has a cause"). With the concept of something that happens, we can extract the concept of 'an existence that was preceded by a time, etc.', but at no point will we be able to extract from it the concept of cause. The idea of a cause does not belong to the idea of a thing that happens in general. What is it that allows us to make this proposition a priori, then? It cannot be experience, since (as has already been established) experience only provides particulars, without the necessity that a priori judgements have. This question (of how synthetic a priori statements are possible) is central to the critique of pure reason.

V. Synthetic a priori judgements are contained as principles in all theoretical sciences of reason.

1. Statements of mathematics are always synthetic a priori; for example, in the sum '7 + 5 = 12', the number 12 is necessarily the result but isn't implicit in the concept of merely pairing the numbers 7 and 5. No elaboration on or clarification of this pair of numbers will extract the number 12; the statement is therefore ampliative (i.e. synthetic) and, obviously, necessary (i.e. a priori).

2. The natural sciences contain synthetic a priori statements amongst their principles. For example, physics holds as a principle that during all changes of the material world, the quantity of matter remains the same. This is both necessary and ampliative, since there is no clarification of the concept of matter that will reach the concept of persistence. The principle is therefore synthetic a priori.

3. Metaphysics itself, even though it has previously not made successful progress as a science, is supposed to contain synthetic a priori statements, since it isn't concerned with merely taking the concepts that our cognition already provides and creating clarifications of them; it is instead oriented towards amplifying these concepts to a point that is further than experience can go.

VI. The general problem of pure reason.

The primary task of the critique of pure reason is made much lighter by the fact that it can be generally expressed in one question: "How are synthetic judgements a priori possible?" The status of metaphysics depends on this question; the historically consistent failure of metaphysics has perhaps been due to its lack of engagement with this question. David Hume got the closest to such an investigation, but effectively failed when he refused to generalise it and focused only on statements that connect cause with effect; his result was to conclude that these statements are in fact not a priori and only appear to be as the result of an illusion after mere habit had created the concepts of cause and effect merely by borrowing from experience. If he had taken his conclusion further, he would have had to conclude that iepure mathematics is also impossible, since (as has been shown) it contains synthetic a priori statements.

Contained within our question regarding the synthetic a priori are the questions regarding the grounding of many applications of reason: for example, How is pure mathematics possible? and How are pure natural sciences possible? Also, regardless of whether or not it becomes established as a science, we cannot deny that the questions of metaphysics arise from a certain natural predisposition in the mind. The critique will therefore also ask: How is metaphysics as a natural predisposition possible? Beyond this, however, we have to examine the objects of the questions of metaphysics, and whether or not they actually lie within the bounds of reason. The final question of the critique is therefore: How is metaphysics possible as science?

The science that the critique will establish will not be 'extensive', since it will be concerned not with the objects of metaphysical questions, but with the questions themselves. Any other attempt would be dogmatic, and dogmatic use of reason leads to groundlessness, which leads to skepticism. With the critique of pure reason, all previous (i.e. dogmatic) attempts at metaphysics are undone.

VII. The idea and division of a special science under the name of a critique of pure reason.

Reason provides principles of all cognition a priori. An organon of pure reason would be a complete list of all these principles. This, however, might not be possible; instead, then, the Critique resorts to a propaedeutic of pure reason, which examines the sources and boundaries of such principles. This is not doctrine but critique, since it aims not to amplify our reason but to purify it and kepe it free of errors.

All cognition that is concerned not with objects but with our mode of a priori cognition of objects is called transcendental. A system of concepts built from such cognition is called transcendental philosophy. But any such system would be too broad for the mere propaedeutic that the Critique hopes to achieve, since we are only concerned with one kind of cognition - synthetic a priori. This investigation serves as the preparation for an organon. This should be possible, since we are not concerened with the (inexhaustible) nature of things, but with the finite a priori capacity of our finite understanding.

We also shouldn't expect a critique of historical books and systems of reason, since such a critique can only hope to find any grounding following a critique of the understanding, prior to any (groundless) systematisation.

Transcendental philosophy is the system of the principles of pure reason. The critique of pure reason forms an architectonic plan for such a system. As such, the Critique presents the 'complete idea of transcendental philosophy', but 'not yet this science itself'.

The most important distinction of transcendental philosophy is that it must contain zero concepts that are in any way empirical - its cognitions must be purely a priori. As such, principles of morality do not enter the science, because they consider the concept of incentive.

The Critique is split into two parts: a Doctrine of Elements and a Doctrine of Method. All that is left to be said in terms of an introduction is this: human cognition has two stems - the sensibility, through which objects are given to us, and the understanding, through which they are thought. The first part of the doctrine of elements will examine if the sensibility has a priori representations that determine how objects are given to us.

Transcendental Doctrine of Elements, first part: The Transcendental Aesthetic

Cognition is immediately directed at object through intuition. All thought as a means is directed to the end of intuition. However, this is only possible if the object is given to us and if it affects the mind in a certain way. Our capacity to receive objects is called the sensibility, and it is the sensibility that give us intuitions. These are processed in thought by the understanding, which gives rise to concepts. All thought must ultimately be related to intuitions, whether directly or indirectly, since there is no other way that objects can be given to us.

The effect an object has on our capacity for representation is called sensation. Intuitions which are related to their objects through sensation are called empirical. The undetermined objects of empirical intuitions are called appearances.

In an appearance, that which corresponds to its sensation is called the matter; the particular ordering of the manifold appearance is called form. Form, being that which orders sensations, cannot itself be given in sensation; as such, it must lie in the mind a priori; sensation, on the other hand, is a posteriori.

A representation is pure if it contains nothing derived from sensation. One such pure representation is the form of sensible intuitions as such. This is encountered in the mind a priori, and is also referred to as pure intuition. If I take a representation, and subtract from it everything that the understanding thinks about it (substance, force, divisibility, and so on), and then subtract everything that comes from sensation (hardness, colour, texture, and so on), that which is left comes from pure intuition. This remainder, namely, is extension and form. These come a priori from the mind, rather than coming from an object given in sensation.

The transcendental aesthetic is the science of the principles of a priori sensibility. The science of the principles of pure thinking is the transcendental logic.

In the transcendental aesthetic, everything that the understanding thinks about representations will be pushed aside, and then everything that is given to representations through sensation will be pushed aside; all that remains will be that which is provided through pure intuition, the mere form of appearances in general. The investigation will find that there are two pure forms of intuition: space and time.

First Section: On Space

Our mind possesses 'outer sense' which allows objects to be represented to us as outside of us, in space. Their size, position, shape, and so on, is all determined in space. The mind also possesses 'inner sense', which does not give us a intuition of the soul itself, but provides an intuition of the mind's inner state. Because of this inner sense, everything that is determined within the mind is represented as being in time. Therefore: space is always and only intuited as something outside us; time is always and only intuited as something within us.

What are space and time? They might be actual entities; they might be merely relations between things, that would exist even without the things being intuited; they may alternatively be relations attached to intuitions, therefore belong only to the constitution of the mind, and not to the things themselves. To answer this question, we will first consider space, by representing (even if not completely) that which belongs to the concept of space.

1. Space is not an empirical concept drawn from outer experiences. Because, in order for something to be represented as 'outside' me, I must already have a representation of space to serve as the ground of this outsideness. In other words, space cannot come from outer experiences, because 'outer experience' is only possible through a representation of space.
2. As established above, space is an a priori representation that provides the ground for outer experience. It is therefore necessary: I cannot represent the non-existence of space, and all appearances in general have space as the condition of their possibility.
3. Space is not the general concept of things' relations in general; it is a pure intuition. The mind can only represent one single space that may be divided into parts - it cannot represent multiple spaces. The parts that space can be divided into cannot precede the single, overarching space, because they must necessarily be represented as being in the latter. It follows, then, that space is grounded in an a priori intuition. For this reason, all geometric truths are derived not from general empirical concepts, but are instead arrived at with apodictic certainty through a priori certainty.
4. The original representation of space is not a concept. Space is represented as an infinite given magnitude. Admittedly, concepts are necessarily thought of as individual representations under the presumably infinite set of all concepts, but no concept itself is thought of as containing infinite representations. Therefore space, having infinite magnitude, cannot be thought of as a concept, but must be thought of as an a priori intuition. Note: I have emphasised 'original representation' because I think it is particularly significant to this passage. On the surface, Kant seems to be denying that space is a concept, only to subsequently write about it as a concept (see the heading below). I don't think he has actually made this mistake; I think that instead, he only argues that the original representation of space - that is, the initial way it is represented to consciousness, and the way from which all (potentially conceptual) cognition can be built - is non-conceptual (or, perhaps, 'pre-conceptual').

Transcendental Exposition of the Concept of Space.

A transcendental exposition is an explanation of a concept as a principle that allows certain synthetic a priori cognitions to be made. Such an explanation can only be carried out if this synthetic cognitions genuinely stem from the principle in focus, and if they are only possible on the condition of the principle.

Geometry is a science that determines particular properties of space. It does so a priori, and its statements are ampliative. As such, the representation of space that geometry makes use of must be an intuition, and not a concept, since concepts can only be analysed, without any ampliative results. This intuition of space must also be a priori, since the cognitions of geometry for which it serves as the condition of possibility are necessary or apodictic.

How can an outer intuition like space (see here) be a priori? The only answer is that space is an immediate representation in the mind, as part of the mind's constitution for receiving intuitions of objects. Only this answer explains the possibility of synthetic a priori cognitions regarding space (i.e. cognitions of geometry, in this case).

Conclusions from the above concepts.

• Space is not merely a property of objects and nor is it merely a relation of things to one another, since such properties and relations can only be thought of after the intuition of an object, and therefore cannot be intuited a priori; yet we do make cognitions of space a priori.
• Space is nothing but the form of all outer intuition - it is therefore the condition of possibility for sensibility as such. We can only speak of it from a 'human standpoint' - that is, we can only speak of it with regard to how it determines our reception of objects in outer intuition. Space must be understood as a condition of possibility for appearances of objects, and not of the objects themselves. The proposition 'all things are next to each other in space' is only true with the limitation that 'things' designates things as they are represented to us. The proposition 'all things, as outer intuitions, are next to each other in space' is universally true. We can therefore say that space is empirically real (i.e. it is real with respect to all outer intuition) but transcendentally ideal: it is nothing as soon as we leave the scope of objects represented as outer intuitions. Outer objects in themselves aren't known to us at all; we are only aware of them as they are represented to us through the 'lens' of such a priori intuitions like space.

Second Section: On Time

Metaphysical exposition of the concept of time.

1. Time is not an empirical concept drawn from experience, because the very possibility of perceiving multiple things as existing simultaneously or in sequence relies upon their grounding in the concept of time a priori.
2. Time is a necessary representation that serves as the condition of possibility for all representations. You cannot remove from an appearance the fact that the appearing object exists in time.
3. The necessity of time is also the condition of possibility for necessary and a priori statements to be made about time. For example: that different times are successive and not simultaneous (in the same way that different spaces are simultaneous and not successive). Such principles aren't derived from experience; if they were, they wouldn't hold necessity.
4. Time is not a discursive concept (i.e. a concept derived merely from the relations between things); instead, it is a pure form of sensible intuition. Different times are simply parts of one single time. All necessary principles regarding time (as described above) are contained inherently within time as a pure intuition.
5. The original representation of time is infinite and unlimited; every limited amount of time is grounded on a limitation of the single, unlimited time that is already intuited.

Transcendental exposition of the concept of time.

The concepts of alteration and motion are only possible through the representation of time. If the mind possessed no a priori inner intuition of time, then things could not be represented as undergoing alteration or motion.

Conclusions from these concepts.

• Time is not something that is simply attached to things as an additional determination; it is the grounding of the representations of these things in the first place. If it were just an objective determination to be appended to representations, then we could represent 'time' without any object, and this is not possible.
• Time is merely the form of inner sense - it is not something that determines outer appearances, since it clearly has no shape or position. Instead, it is that which determines the relation of representations in the mind's inner state. We also know that the representation of time is an intuition since it can be expressed in an outer intuition - for instance, when we picture time as a two-dimensional line stretching into the past and future.
• Time is the condition of possibility for all representations. Space, on the one hand, only determines the representation of outer things, since objects represented as within the mind don't take place in space. However, all objects - whether they are 'inner' or 'outer' - take place in time and are therefore grounded upon the a priori intuition of time. To be precise, we should say not that 'all things are in time', but that 'all things as appearances (or objects of sensible intuition) are in time'. The latter statement is a priori and therefore universal and objective. Just like space, then, time is empirically real, meaning that it is objectively real and valid with regard to objects given to our senses and represented to us; and it is transcendentally ideal, being reduced to nothing if we abstract from subjective sensible intuition.

Elucidation.

There is an objection to the claim that time is empirically real but only transcendentally ideal. The objection claims that alterations are real - which is proved by the change of our representations - and that alterations can only occur within time; as such, time must be real. Kant simply concedes to this criticism: yes, time is something real - it is the real form of inner intuition. It shouldn't be regarded as an object, but as the way I represent myself to myself as an object. This is the empirical reality that Kant ascribes to time; it is only absolute reality that it doesn't possess.

Time and space are two sources of cognition, and they are the pure forms of all sensible intuition. This is how they grant the possibility of synthetic a priori propositions. Time and space, as sources of cognition, are limited by the fact that they are conditions only of sensibility and not of things; as such, the cognition they enable applies only to things as appearances, rather than things as they are in themselves.

The transcendental aesthetic only contains space and time, since any other concepts that belong to sensibility can be proved to have empirical presuppositions. For example, even a concept so simple as motion is empirically sourced, since it relies not only on the a priori intuition of space, but also on an empirical observation of something movable - that is, an observation made through experience. Space itself (as an a priori intuition) does not contain the concept of movement.

General remarks on the transcendental aesthetic.

1. All intuition is nothing but the representation of appearance. We do not represent things as they are in themselves; if we remove all the subjective constitution of the sensibility from our representations, all relations of things in space and time - and even space and time themselves - would vanish. Whatever the case is with these objects in themselves is 'entirely unknown to us'.

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   Space and time are the pure forms of our perception, and sensation as such is its content. The former are provided a priori, and as such are known as pure intuitions; the latter is what is responsible for all a posteriori cognition (i.e. empirical intuition).

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   Sensible knowledge and conceptual knowledge are of different kind and not just of different quality, as Leibniz claimed (arguing that 'our entire sensibility is nothing but the confused representation of things'). Knowledge of the sensibility is always particular; conceptual knowledge is always universal.

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   The transcendental aesthetic isn't just a plausible hypothesis; it holds as much certainty as we can expect from a theory. This certainty is evident from the fact that we have synthetic a priori knowledge: synthetic knowledge cannot be derived only from concepts, since concepts can only be further elucidated to produce analytic (tautological) statements. As such, synthetic knowledge must make use of an intuition. But if this intuition were sensible - given to us a posteriori - then our synthetic knowledge wouldn't hold necessity (and our synthetic a priori knowledge of geometry, for example, does). As such, in making synthetic a priori cognitions, we rely upon a priori intuitions - space and time - which condition our representations of objects.

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2. Everything in our cognition that belongs to space and time as intuitions is built up entirely from relations: of places in one intuition (extension), of the alteration of places (motion), and of laws which determine this alteration (moving forces). But what is actually in the place - that is, the content of a sensation - is not given by the intuitions of time and space which, together, provide only the forms of intuition. The forms of intuition order sensations, but concepts are needed to unify (or 'particularise') these sensations.

   "Now through mere relations no thing in itself is cognized; it is therefore right to judge that since nothing is given to us through outer sense except mere representations of relation, outer sense can also contain in its representation only the relation of an object to the subject, and not that which is internal to the object in itself." (A49/B67)

   Space (and time) is a relational property of things; it is therefore not internal to the things, and shouldn't be considered as a property of the things in themselves. It is simply assumed that relational properties aren't 'internal to the object': for instance, water has the internal property of being comprised of hydrogen and oxygen. It has the relational property of being in a glass; and, if it is taken out of this glass (and loses the relational property), it is still (presumably) the same thing. The relational properties of the water are therefore not internal to the water as an object. As such, everything that is represented to us after being taken in through the senses is only appearance. In conclusion: outer sense provides us only with appearances and not with things as they are in themselves.

   ---

   The same is true with inner sense (of which time is the form - see above). Inner sense grounds the way in which representations are placed in our mind, by ordering them in time - allowing us to be conscious of representations occuring simultaneously, or in sequence, and so on. It therefore (like outer sense / space) contains nothing but relations; it is therefore a form of intuition. Inner sense represents nothing other than how things are posited in the mind, and it is therefore 'the way in which the mind is affected by its own activity'. Since, as it was shown above, everything represented through (inner or outer) sense is merely appearance; it follows that the mind, through inner sense, is revealed to us only as appearance. Inner sense allows me to be conscious of my representations, in their determined order, but it does not make me aware of my self as the subject of my representations.

   ---

3. The claim that intuition represents outer objects and the self-intuition of the mind, in space and time, as appearances, does not equate to a claim that these objects are just illusions. In any appearance, the objects always appear to be something genuinely given The same is true for the qualities/properties we attribute to these objects. Any such qualified object (i.e. any such object with properties), insofar as its appearance is dependent on the subject's way of intuiting it, is distinguished as appearance from however it may be in itself. If we assumed instead that space and time were properties of things in themselves then, given that they are infinite, the absurdities that would arise would lead us to claim all bodies to be ideal, as Berkeley did.

---

4. As universal as the intuition of space is to finite objects, it remains an intuition regarding the senses, and cannot therefore be ascribed to supposed infinite beings, such as a god. As Kant writes in the Inagural Dissertation:

   "What, however, in immaterial substances constitutes the external relations of force between them or between them and bodies, obviously eludes the human intellect. … But when men reach the conception of a highest and extra-mundane Being, words cannot describe the extent to which they are deluded by these shades that flit before the mind. They picture God as present in a place: they entangle Him in the world where He is supposed to fill all space at once. They hope to make up for the limitation they thus impose by thinking of God's place per eminentiam, i.e. as infinite. But to be present in different places at the same time is absolutely impossible … what is in several places is outside itself, and is therefore present to itself outside itself - which is a contradiction in terms."

Conclusion of the Transcendental Aesthetic.

The first question of transcendental philosophy - namely: how are synthetic a priori propositions possible? - is now closer to a solution. Synthetic a priori propositions are possible through a priori intuitions - space and time - which allow us to go beyond the mere analysis of concepts in a priori judgments and make synthetic claims, not by analysing the concept but by exploring the intuition that corresponds to it. As such, however, such judgments will never go beyond objects of the senses that belong to possible experience.

Transcendental Doctrine of Elements, second part: The Transcendental Logic

Introduction: The Idea of a Transcendental Logic

I. On logic in general.

All cognition arises from two sources in the mind: (1) the reception of representations (or impressions), and (2) the faculty for cognizing an object through these representations (the 'spontaneity of concepts'). The first gives us an object, and through the latter the object is thought. The elements of all cognition, therefore, are intuition and concepts. Concepts without intuitions filling them cannot provide cognition, just as intuitions without concepts to make sense of them can provide cognition. Such cognition can be either empirical (if sensation is contained within it - that is, if its object is actually present) or pure (if no sensation is contained within it). Sensation is the 'matter' of sensible cognition; as such, pure intuition contains only the 'form' of how things are intuited. Pure concepts are simply the form of thinking an object in general. Empirical intuitions and concepts are never possible a priori.

The above paragraph establishes that cognition arises from the reception of representations, and the subsequent cognizing of a represented object - in other words, the thinking of the object. The former stage is carried out by the faculty named sensibility, the latter through the faculty named understanding.

Intuition is always sensible, since it contains only the way in which the mind is affected by an object. But the thinking of any intuition always takes place in the understanding. Both the sensibility and the understanding are necessary, and neither one has privilege over the other, since without the sensibility, the understanding would have no object given to it; and without the understanding, the sensibility would have its objects left unthought. Concepts must be sensible and intuitions must be understandable.

"Thoughts without content are empty, intuitions without concepts are blind." (A51/B75)

The Transcendental Aesthetic concerned the rules of sensibility in general. The Transcendental Logic concerns the rules of the understanding in general. This logic can itself be divided into two parts, concerned with the rules of either the general use of logic, or the particular use of it. The former category is 'elementary logic' and contains the necessary rules under which all thinking is conducted, without regard for the kind of object being thought; the latter is concerned with different kinds of objects, and therefore should be understood as the organon of any particular science that corresponds to a particular kind of object.

Further classification can be made: general logic (the kind concerned with the universal, object-agnostic rules of thinking) is either pure or applied. Pure general logic abstracts from all empirical conditions - i.e. the influence of the sense - that our understanding is typically exercised on. It therefore abstracts from memory, habit, inclination, prejudice, the imagination, and so on. As such, pure and general logic only concerns itself with the a priori principles of thinking. It is therefore a canon of the understanding (and of reason), regardless of the understanding's potential content.

On the other hand, general reason is applied when it concerns the uses of the understanding under empirical conditions. Nonetheless, it still takes no concern of the different kinds of empirical objects, since it is still general and therefore concerned with only necessary rules. Applied general reason is therefore neither a canon of the understanding as such, nor an organon of any particular science. It is 'merely a cathartic of the common understanding'.

The present science (i.e. the critique of pure reason) will be concerned with pure general reason, since that alone can be examined in the process of working out the doctrine of the elements of the understanding. Two things must be considered in this examination:

1. Since we are concerned with general logic, we will not be concerned with the content of cognition, and will abstract away from the difference of objects, focusing only on the form of thinking.
2. Since we are concerned with pure logic, we will consider no empirical (i.e. psychological) principles, and will arrive only at a priori (and therefore necessary) aspects of a doctrine.

Again, applied logic is not the focus of the Transcendental Logic; the relationship between pure logic and applied logic is analogous to the relationship between the laws of morality and the concrete ways in which passions, feelings, and inclinations affect moral decision-making.

II. On transcendental logic.

(Note: from this point onwards, 'general logic' refers to 'pure general logic' unless I state otherwise.)

General logic abstracts from the content of cognition - i.e. the relation of cognition to the object - and examines only the form of the relations of cognitions to one another, i.e. the form of thinking as such. It is also indifferent to the source of cognition (i.e. whether it is from an empirical intuition or a pure intuition), since it is only concerned with the a priori rules of the understanding that thinks objects, regardless of whether those objects are themselves a priori or not.

Since we are examining transcendental logic, it needs to be clear which cognitions are transcendental and which are not. Not every a priori cognition is transcendental. Transcendental cognitions are a priori cognitions which we use to 'cognize that and how certain representations (intuitions or concepts) are applied entirely a priori, or are possible (i.e. the possibility of cognition or its use a priori)'.

Since transcendental cognition is possible, we can expect that there are concepts which correspond to objects a priori, and that are therefore not of empirical or aesthetic origin. The Transcendental Logic is the science of the cognitions that use such concepts; it is the science that determines the origin and validity of cognitions by mean of which we think objects a priori. The Transcendental Logic is concerned with the laws of the understanding insofar as they relate to objects a priori.

III. On the division of general logic into analytic and dialectic.

A classic question which always provides a challenge to logicians is: What is Truth? Kant presupposes the 'nominal definition' of truth - namely, that it is the correspondence of a cognition with its object - but nonetheless, it can still be asked: what is the truth of a cognition?

If truth is the correspondence (or 'agreement') of a cognition with its object, then the object must be distinct from others, since a cognition isn't true if it agrees with some object, but only the particular object it is related with. We can say that one universal criterion of truth is that it belongs only to cognitions whose form is valid: that is to say, cognitions which express something that is either true or false (i.e. cognitions which are truth-apt). But this criterion only concerns the form of cognitions; what about their content or 'matter'? Well, content - that is, actual things - cannot themselves have a priori criteria for truth, since no empirical object is known a priori. As such, Kant concludes, 'no general sign of the truth of the matter of cognition can be demanded, because it is self-contradictory'.

However, since we nonetheless can have criteria of truth when concerned with the mere form of cognition, we can expect these criteria to be presented in the rules of the understanding that our general logic outlines. Since any cognition that contradicts the rules of the understanding is false, because any such cognition would be the result of the understanding contradicting its own rules, and thereby contradicting itself. It is important to note that, since these rules serve only as truth-criteria for the form of cognition, they are entirely correct and necessary, but not sufficient for the truth of a cognition. After all, a cognition can still be true in form (i.e. not contradictory), but still contradict its object. Logic will not attempt to go further than the formal (i.e. logical) criterion of truth: that is, the agreement of a cognition with the formal laws of reason and the understanding.

General logic contains an analytic, which analyses the faculties of reason and the understanding into their elements, which are the principles of the logical evaluation of our cognition. This establishes a negative criterion of truth: i.e. "a cognition is true if it is not …". To establish a positive criterion of truth for a cognition, we would have to investigate its content; since its positive truth relies upon the relation to its object rather than just its mere form. Mere logic cannot say anything about the objects of cognition.

Note: an example of a negative criterion for the truth of a cognition is the principle of non-contradiction. A cognition is true if it doesn't contradict itself.

Despite the fact that logic may only provide negative criteria for the truth of cognitions, there have nonetheless been philosophers (e.g. Wolff, Leibniz) who have attempted to transform these criteria into positive criteria and go on to construct metaphysical systems out of this transformation. This error is equivalent to taking logic to be a organon for the production of assertions, rather than merely a canon for judging them. Logic as canon is called analytic; logic as organon is called dialectic.

The philosophers who fell into the trappings of dialectical logic only provided 'the air of truth', by imitating practice that is thorough and in accordance with the rules of logic. When general logic is taken as an organon, it is always a 'logic of illusion'. It teaches us nothing about cognitions and their content, but only 'the formal conditions of agreement with the understanding' - it is 'idle chatter' that always forgets to consider the content of cognitions and, without any authentic content grounding it, falls into pretension.

IV. On the division of transcendental logic into the transcendental analytic and dialectic.

The transcendental aesthetic isolated the sensibility and considered only those aspects of cognition which came from the sensibility. The transcendental logic does the same, but with the understanding instead of the sensibility. We thus arrive at an examination of pure cognition. The use of such cognition depends upon the fact that there are objects given to us through intuition to which pure cognition can be applied. Without intuition, cognitions would lack objects and would therefore be empty.

The transcendental analytic presents the elements of the understanding's pure cognition, and the principles without which no object can be thought at all. The transcendental analytic is therefore also a logic of truth: no cognition can contradict its rules without entirely losing its content (i.e. its relation to any object) and therefore becoming false.

It is tempting to go beyond the actual use of the understanding's pure cognitions, and attempt to make speculations with them without providing them with content within the bounds of possible experience. To do so is to use logic as an organon and not merely a canon for the empirical use of the understanding. The second part of the transcendental logic is named the transcendental dialectic and is dedicated to a critique of this erroneous, dialectical and 'hyperphysical' use of the understanding.

Transcendental Logic, First Division: The Transcendental Analytic

The transcendental analytic will analyse all our a priori cognition and break it down into the elements of the understanding's pure cognition. It maintains the following concerns throughout:

1. Its concepts will be pure, rather than empirical
2. Its concepts will belong to thinking and the understanding rather than intuition and the sensibility
3. Its concepts will be elementary, distinguished from those which are derived or composed from elementary concepts
4. The table of its concepts will be complete, making up an exhaustive account of pure understanding

This list will not be satisfied by the aggregation of mere estimates; as such, the transcendental analytic will begin with an idea of the whole of a priori cognition and will establish, through the division of concepts, each concept's connection to the others in a system. The pure understanding is separated from all sensibility and all things empirical; it is a systematic unity that can stand on its own. Taken all together, then, each cognition within it is validated by all the others.

The transcendental analytic will be in two parts; the first concerns the concepts of the pure understanding, and the latter concerns its principles.

Transcendental Analytic, First Book: The Analytic of Concepts

The 'analytic of concepts' that takes place in this section isn't the typical philosophical procedure of analysing concepts for the sake of increasing their distinctness and clarity; rather, it is an analysis of the faculty of the understanding itself, with the aim of finding out the possibility of a priori concepts which originate in the understanding. These concepts 'lie ready' in the understanding until experience is received by the mind and the concepts are brought to use.

First Chapter of the Analytic of Concepts: On the Clue to the Discovery of all Pure Concepts of the Understanding

(i.e. The Metaphysical Deduction)

As the faculties of cognition are used, various concepts will arise which, as they are brought to attention more often and thereby become more clarified, can be collected in a reasonably exhaustive set. This way of discovering concepts won't grasp them in any ordered or systematic way; however, the concepts can be grouped according to their similarity, and they can be manually ordered on the basis of their magnitude: 'from the simple to the more composite'. These ordered groups are not systematic, though they are methodically produced.

In seeking its concepts, transcendental philosophy must be conducted in accordance with some principle, so that the place of each pure concept, and the completeness of their totality, can be determined a priori.

First Section: On the logical use of the understanding in general.

The understanding has above been described negatively: it is a non-sensible faculty of cognition. It is therefore not a faculty of intuition, since intuition cannot occur without sensibility. The only other way for cognition to occur is through concepts and, as such, this is the way that the understanding works. It is discursive rather than intuitive. While intuitions are based on 'affections'; concepts are based on functions. A function is the act of ordering different representations under a common one. Concepts therefore work with the spontaneity of thinking, whereas intuitions work with the receptivity of impressions. The only use of concepts is making judgments with them, since only intuitions relate immediately to an object. A concept has no immediate relation with any particular object; instead, it is related to some representation of an object (whether that is an intuition, or another concept).

Kant gives an example of a judgment: "All bodies are divisible". This judgment contains the concept 'divisibility', which itself is related to the concept of 'body'. The concept of 'body' relates to certain possible intuitions that we receive; this is how the judgment is 'grounded' and actually refers to something. As this example shows, concepts allow for the unification of certain representations/intuitions. In a judgment such as this example, multiple possible cognitions concerning particular representations are united in a cognition concerning a higher, conceptual representation. This judgment is made possible by the faculty of the understanding - as all judgments are.

The understanding is therefore a faculty for thinking, and 'thinking' simply means cognition through concepts. The concepts involved in thoughts can be understood as predicates for possible judgments.

Second Section: On the logical function of the understanding in judgments.

If we take any judgment, and abstract from its content until all we have left is its form, which is provided by the understanding, then we find that its fundamental functions can be brought under one of four titles, each of which contains three different parts. This table represents this grouping of functions of the understanding (I have coupled each part with an example judgment):

A note: the difference between assertoric and apodictic judgments is that the former assert that something is or is not the case; the latter assert that something is (or is not) the case necessarily.

In some ways, this table differs from the traditional techniques of logic; as such, there are some reasonable concerns and clarifications that Kant raises:

1. The first protest is that singular judgments (e.g. A is B) are equivalent to universal judgments (e.g. All A are B) because, in the former, the predicate applies to all of the subject. However, the relationship of singular judgments to universal ones, with regard to quantity, is the same as the relationship of unity to infinity. Therefore, singular judgments are distinct from universal judgments and deserve their own place in the table.

2. Secondly, while general logic puts infinite judgments in the same category as affirmative ones, transcendental logic keeps them separate. General logic has no concern for the content of the predicate; it only considers whether it is attributed or opposed to the subject. For the concerns of transcendental logic, however, the content or value of the predicate has a bearing on cognition. Thus to claim that "the soul is not mortal" is different from claiming that "the soul is non-mortal"; the latter makes an actual affirmation.

3. There are three forms of relation in the table of judgments. Categorical relations are relations between subject and predicate; hypothetical relations are between the ground and the consequence; disjunctive relations are between a divisible cognition and all the members of its division. As such, the first two forms of relation can only relate two concepts (e.g. the proposition "if there is perfect justice, then obstinate evil will be punished" links two propositions - the one before "then" and the one after). In disjunctive relations, however, there can be two or more propositions involved in the relation: we could have, for example, the proposition: "The world exists either through blind chance, or through inner necessity, or through an external cause". Note: During Kant's time (and generally during the time before Frege's work), disjunctions in formal logic were typically taken to be exclusive. As such, Kant describes disjunctive judgments as follows:

"In a disjunctive judgment there is therefore a certain community of cognitions, consisting in the fact that they mutually exclude each other, yet thereby determine the true cognition in its entirety, since taken together they constitute the entire content of a particular given cognition." (A74/B99)

4. The function of modality is unique, in that it contributes nothing to the content of the judgment (the content of a judgment is nothing more than its quantity, quality, and relation). Instead, modality concerns the 'value of the copula in relation to thinking in general'. Problematic judgments concern assertions or denials that are only possible; assertoric judgments concern assertions or denials that are actual (that is to say, actually true); apodictic judgments are those which hold assertions or denials to be necessary. As such, any hypothetical judgment ("If A then B") contains two propositions ("A" and "B") that are merely problematic. The three forms of modality can also be understood as 'so many moments of thinking in general'; since thought so often involves something being judged problematically, before it is assumed assertorically as true, and then finally established as apodictic.

Third Section: On the pure concepts of the understanding or categories.

Let's remember that our question is how are synthetic a priori judgments possible? The first step towards an answer was completed by the Transcendental Aesthetic, which proved the existence of space and time as a priori intuitions that make experience possible. The second step, explored here, consists in demonstrating that any such (synthetic a priori) judgments about objects of experience are given conceptual content only if (1) certain categories of the understanding order our representations (of these objects), so that we can (2) form concepts about them, and (3) use these concepts in judgments.

This section will arrive at a table of these categories. The following section (the transcendental deduction) will prove their objective validity. While Kant's predecessors, in the tradition of Aristotle, tried to give a priori arguments for the categorical, universal features of things in themselves, the metaphysical deduction only argues that our understanding is so structured that it can only arrive at objective representations of appearances by using the categories that the deduction discovers.

A reminder of the difference between intuitions and concepts. Intuitions are singular, meaning that they are the representatives of individual things; they are immediate, not requiring the mediation of other representations in order to relate to an object. Concepts, on the other hand, are general, representing not particular objects but properties that can be common to several objects; they are mediate, meaning that they can only relate to individual objects through the mediation of other representations (i.e. intuitions).

In the Transcendental Aesthetic, Kant demonstrated that space and time are a priori intuitions; they are forms of sensibility through which things given to us by the senses are related to one another. Now, Kant will demonstrate that we also have concepts innate to the mind which apply universally to objects. These include existence, cause, necessity, possibility, and substance. This discovery raises the question: how can concepts that originate in our minds be applied to objects that are given? Empirical objects and our concepts of them are supposed to be entirely independent; how can the mind then bring them together?

In answering this question, Kant distinguishes the two ways in which the mind grasps reality: through receptivity and through spontaneity. The latter is the power of the sensibility - it is what allows the mind to be affected and thereby receive representations; the latter is the understanding's capacity to think over these representations and form concepts from them.

Both the mind's receptivity and its spontaneity have their own forms of representation: the former provides intuitions, and the latter produces concepts. Concepts are related to one another in judgments and inferences, and all intuitions are themselves ordered within the single intuition of space and the single intuition of time, both of which are a priori:

"Now space and time contain a manifold of pure a priori intuition, but belong nevertheless among the conditions of the receptivity of our mind, under which alone it can receive representations of objects, and thus they must always also affect the concept of these objects." (A77/B102)

If the first two sections of this chapter established the idea that the understanding is a capacity to make judgments, this third section defends the thesis that all judgments presuppose synthesis.

In some sense, this idea has been defended since Aristotle, who established the fact that any judgment "A is B" involves a synthesis of the concept "A" with the concept "B". But the Kantian notion of synthesis is separate from this. The synthesis that is focused on here is that which allows concepts of sensible objects to combined into judgments at all. Insofar as we are concerned with concepts of sensible objects, judgments can only be made by synthesising these concepts if their objects are themselves synthesised (and not just left as a manifold of parts). Transcendental logic is concerned with this synthesis of intuited manifolds into distinct objects.

The reason such a synthesis is required in the first place is that, in order for analysis of intuitions into concepts to take place, those intuitions, which first appear as manifold, must first be synthesised.

In judgments that concern a priori manifold intuitions, synthesis can take place simply through a priori concepts. For example, in geometrical judgments, the concept of a 'triangle' is immediately presented as a synthesised thing, because it is a merely spatial manifold, and space is an a priori intuition. In metaphysical judgments, however, our objects are independent of us and, as such, we do not come equipped with a priori concepts for them.

How, then, do we construct representations of these independent objects? It is not the way we construct mathematical representations, since the guidelines for that process are given a priori. Instead, our mind receives a manifold of intuition, and unifies it according to particular rules. After this unification, analysis of the intuition can be made into concepts which can then be thought in judgments.

Kant claims that "to bring this synthesis to concepts is a function that pertains to the understanding and by means of which it first provides cognition in the proper sense" (A78 / B103). The idea of bringing synthesis to concepts can be illustrated as such: everything that is given to us in sensibility is disperse and in disunity. Two similar sensible things (e.g. two trees) couldn't be compared unless we grouped the intuitions that make them up (i.e. intuition of leaves, a trunk, and so on) according to regular patterns. But what determines which patterns are applied (i.e. why is it that the concept of "tree" is leaves, branches, and trunk? Why not just branches and trunk, leaving the leaves as a separate 'thing'?). The regularity of our sensible concepts implies that the rules of synthesising these manifolds must therefore by systematised in the understanding. This is the understanding's bringing of "synthesis to concepts". In other words, the understanding provides a ground of unity that allows synthesis to function. This ground of unity is always a pure concept of the understanding.

As an example, Kant shows how we have the ability to count not only in digits, but in tens, hundreds, and so on, thereby showing our ability to synthesise collections of individual objects. This process of counting can be applied to empirical objects (for example, when measuring the length of a line), showing our ability to synthesise sensible manifolds.

It is at this point that Kant makes one of his most genius points so far:

The same function that gives unity to the different representations in a judgment also gives unity to the mere synthesis of different representations in an intuition, which, expressed generally, is called the pure concept of the understanding. The same understanding, therefore, and indeed by means of the very same actions through which it brings the logical form of a judgment into concepts by means of the analytical unity, also brings a transcendental content into its representations by means of the synthetic unity of the manifold in intuition in general, on account of which they are called pure concepts of the understanding that pertain to object a priori; this can never be accomplished by universal logic."

This paragraph is incredibly dense, but it seems to claim that if our judgments have certain necessary forms (forms that the table of judgments enumerated), then the concepts of the objects involved in those judgments must also be structured in a specific necessary way, so that we can use them in these judgments. If this is the case, then our intuitions (which are the source of those objects, and the ultimate target of judgments) must also be structured in certain necessary ways.

For instance - if a categorical judgment ("A is/belongs to B") asserts that a particular predicate applies to a particular subject, then any object involved in such a judgment must be understood as a substance [subject] with properties [predicates]; further, we must therefore be able to distinguish substance from property in our intuitions.

The categories should therefore be understood as concepts of objects in general, which determine how those objects function in a judgment: for instance, the concept of substance determines that any object that falls under such a concept should be understood as the subject of a judgment.

It's important to note that categories do not stand for particular objects: for example, the category of accident (i.e. property) doesn't refer to 'solidity', 'transparency', or any other particular property. It is instead the form of properties in general, just as space and time are forms of intuition. The difference here is that we intuit space and time as singular objects, but the pure concepts that the categories represent are never intuited as particulars.

Kant notes that there are an equal number of categories as there are functions of judgment; this is because "the understanding is completely exhausted and its capacity entirely measured by these functions" (A79 / B105). This doesn't mean that the use of a particular function of judgment implies a use of its corresponding category; a category only 'arises' when the understanding's judging capacity is applied to sensible manifolds - these manifolds become synthesised in intuition, analysed into concepts, and again synthesised into judgments.

One thing to note in the table of categories is that, in each class, the third category arises from the combination of the first two above it: for example, totality 'is nothing other than plurality considered as a unity' (B111). But these third categories aren't merely derivative; they are still as 'ancestral' as the others, because the act of combining the first and second categories of each class to form the third is a unique act of the understanding.

Second Chapter of the Analytic of Concepts: On the Deduction of the Pure Concepts of the Understanding

(containing the Transcendental Deductions)

In legal theory, jurists distinguish between what is lawful and what is factual. The proof of the former - i.e. the defence of a claim of lawfulness - is known as a deduction. Though we make use of a variety of empirical concepts very regularly, not all of them have a deduction: for example, there is no justification from experience or reason for the use of the concepts fortune and fate.

As it has been demonstrated above, some concepts can be used in a 'pure', a priori way. As such, they lie before experience and cannot be justified with referral to experience; their justification must therefore be deduced with reason. Only following this justification can we know that these concepts, which arise independently of experience can correctly apply to objects of experience. The process of giving this justification is called the transcendental deduction. It is distinct from an empirical deduction, which simply shows how a concept has been acquired from experience, without giving any justification to the concept.

Space and time, along with all the categories, are a priori concepts which apply to objects of experience, and only a transcendental deduction of them is possible. It might be objected that we can do as Locke did, and look to experience to see which kind of impressions give rise to the use of these concepts, thereby going from individual perceptions to general concepts. However, this is not possible for a priori concepts: such a method - a 'physiological derivation' - only explains our possession of these pure a priori concepts; it does not justify them. As such, any deduction of these concepts must be transcendental.

A critic might then object: is a transcendental deduction really the only method available to us? After all, geometry can prove its claims a priori without having to resort to a philosophical justification of its central concept (space). Geometry, however, is different insofar as it is grounded on intuition which is given a priori (the pure intuition of space); as such, both its form and its content are already given. With the pure a priori concepts of the understanding, a transcendental deduction is still needed, because these concepts apply to objects in general, without any conditions of sensibility. They aren't presented with objects from a priori intuitions on which to ground their synthesis, as geomtry is. Furthermore, a transcendental deduction is needed to delimit the valid (i.e. justified) use of these concepts, to prevent their use beyond the boundaries of possible experience.

The Transcendental Analytic straightforwardly gave a deduction of the pure concepts of space and time, because space and time are, evidently, the conditions of possibility for the appearance of objects in general. Deducing the categories, however, is a more difficult task, since objects can be intuited by us without being related to particular categories. The categories do, however, provide the condition of possibility of cognizing objects; we therefore have to ask the difficult question of how subjective cognitions of thinking can apply to objects with objective validity.

We know that objects must comply with the conditions of sensibility, or else they wouldn't be objects for us at all. It is harder to determine why they must, necessarily, also comply with the "conditions that the understanding requires for the synthetic unity of thinking" (A90 / B123). Hypothetically, appearances could come to us without any conformity to the conditions of the understanding, in a state of complete confusion wherein concepts such as cause and effect wouldn't apply and would therefore be empty. In this scenario, the mind would still intuit appearances, but the objects of these appearances would simply be unthinkable.

We cannot do as Hume did and claim that concepts come from habitual generalisation of regular occurrences; if the categories are to have objective validity then they must originate a priori in the understanding; otherwise, they are just fantasies. "To the synthesis of cause and effect there attahces a dignity that can never be expressed empirically" (A91 / B124) - namely, necessity.

Transition to the Transcendental Deduction

There are two ways of synthetic representations joining up with objects:

1. The object alone makes the representation possible: this is an empirical relation, and therefore holds no necessity; the representation isn't possible a priori.
2. The representation alone makes the object possible: 'representation in itself' doesn't bring objects into existence, but is what allows them to be cognized as objects in the first place.

With regard to the second way: there are two ways that an object can be cognized: either through intuitions (the object is given as appearance) or through concept (the object is thought, in correspondence to an intuition). We know that intuition is subject to a priori rules (as the Transcendental Aesthetic established). We must therefore ask if a priori concepts also condition the way that objects are thought as objects, in a way that - like the a priori forms of intuition - doesn't allow objects to be cognized without them.

All experience contains intuitions and concepts of the experienced object, and concepts are therefore a priori conditions of experiential cognition; as such, if the categories have objective validity it must be the case that they are necessary conditions for experience in general, since only through them can objects of experience be thought at all.

The Transcendental Deduction is therefore directed at recognising the categories as a priori conditions of the possibility of experience. Any analysis of the experience in which the categories are encountered will not serve as a deduction, since this would show them only in a state of contingency.

Locke went about in this way, deriving pure concepts from experience; this gave his project such an inconsistent grounding that he then attempted to cognize beyond the limits of possible experience. Hume recognised that, in order to make cognitions of such a sort, the pure concepts would have to be necessary and a priori; however, he failed to establish how they could be so, since he never thought to look to the understanding as their origin. Hume's empirical approach is refuted simply by the fact that we dohave disciplines of a priori certainty, such as pure mathematics.

The paragraph above shows how Locke embodied philosophical "enthusiasm", and Hume embodied "skepticism". The Transcendental Deduction will show how we can safely navigate between these two extremes.

A Summary of Guyer's Account of the Transcendental Deduction (in the First Edition, aka. the A-deduction)

In the A-deduction, Kant argues that there are three elements involved in all experience of objects: (1) apprehension of several intuitions of an object, providing the manifold of intuition; (2) reproduction of earlier items in a manifold as we apprehend later ones, raising the question of whether the earlier intuitions represent the same objects as the later intuitions; (3) recognition of the unity of the manifold under a concept - i.e. recognising that the several intuitions all represent the same object, because our concept of the object necessitates that the object must have the particular properties that the successive intuitions represent it as having.

There are two problems with this argument:

1. The argument makes use of empirical concepts in the cognition of objects; unless we can show that empirical concepts depend on the a priori categories of the understanding, this argument tells us nothing concerning the objective validity of the categories. However, the metaphysical deduction has arguably already shown us that cognition of objects requires the categories, so we can turn to the second and more important problem with the argument.
2. The argument analyses our cognition of objects, but fails to prove that our experiences must be experiences of objects; as such, it fails to prove that the categories have objective validity in applying to all of our experiences. The argument therefore seems to make no further progress than the metaphysical deduction.

Despite these problems, Kant seems to think that the threefold synthesis shows that cognition of objects must involve a priori concepts; he takes the necessity of a concept to unify the manifold of intuition to imply the necessity of a necessary and a priori concept. This is a fallacy, since ordinary empirical concepts can perform this unification (e.g. the concept "table" can unify a manifold of intuitions of a table).

However, in examining where necessary and a priori concepts could come from, Kant introduces the concept of transcendental apperception, defining it as a "unity of consciousness that precedes all data of the intuitions" or a "pure, unchanging consciousness" of the "numerical identity" of myself in all of my various experiences (A107, A113). Transcendental apperception is the principle stating that, with every experience I have, I also know that I am having that experience, and that having such knowledge allows me to know that this particular experience belongs to the same consciousness (i.e. mine) as all my other experiences. I am the self that is "numerically identical" throughout all of my experiences.

The introduction of this principle, which stands on its own, means that we no longer even need the argument of the threefold synthesis. The new question is whether or not the existence of transcendental apperception implies that the categories are necessarily applied to all experiences that transcendental apperception applies to (i.e. to all experiences). If it does, then transcendental apperception is the basis for the proof of the categories' objective validity.

The mission for the deduction, therefore, is to show that the fact of transcendental apperception leads to the objective validity of the categories.

Kant's method of showing this in the A-deduction is to suppose that the unity of consciousness in transcendental apperception is both synthetic and a priori and, as such, depends upon an a priori synthesis:

"Now the unity of the manifold in a subject is synthetic; pure apperception therefore yields a principle of the synthetic unity of the manifold in all possible intuition.
This synthetic unity, however, presupposes a synthesis, or includes it, and if the former is to be necessary a priori then the latter must also be a synthesis a priori." (A117-8)

In other words, when we ascribe a manifold of representations to a single self, we are asserting that each individual representation is connected to the others by virtue of its belonging to this single self, and - since this connection and belonging is not part of the individual representation considered logically by itself - the connection of the representations (i.e. the unity of the manifold) cannot be discovered by mere analysis. For example, nothing in my experience of observing a candle implies that I was previously looking at a book; the concept of my self (which is numerically identical in both experiences) must be added in order to connect the two experiences. Knowledge of the connection is therefore synthetic.

How does Kant show that the unity of the manifold is not only synthetic but also a priori? In order to do so, he must show that the idea of any representation we have being ascribed to a numerically identical self is necessary. At least in the first edition, Kant makes this claim:

"All intuitions are nothing for us and do not in the least concern us if they cannot be taken up into consciousness … and through this alone is cognition possible. We are conscious a priori of the thoroughgoing identity of ourselves with regard to all representations that can ever belong to our consciousness, as a necessary condition of the possibility of all representations. … This principle holds a priori." (A116)

Following this principle, Kant argues that, if we know there is a synthetic unity of our representations before we even know the content of those representations, then each of our representations must have an a priori synthesis before it is empirically synthesised (i.e. before it is placed under an empirical concept like 'candle' or 'book'). This first a priori synthesis is named the productive synthesis of the imagination. Kant argues that, since it is a priori, it must have its own a priori rules. Assuming that all synthesis is ultimately a product of the understanding - and a product of the pure understanding, in the case of the transcendental synthesis of the imagination - the categories (i.e. the rules/conditions of the pure understanding) must be the conditions of the pure synthesis which results in the unity of apperception. As such, we must apply the categories to all of our representations, and they must serve as the pure forms for all concepts; the categories provide a necessary 'affinity' for all of our representations.

A Condensed Form of the Summary Above:

• Transcendental Apperception is the fact that, with each experience I have, I also know that I myself am having that experience. My self is the consciousness that is numerically identical amongst all my experiences.
• If the categories are necessarily applied to all experiences that transcendental apperception applies to, then they apply to all my experiences, and they are therefore objectively valid on the basis of the principle of transcendental apperception.
• Transcendental apperception is synthetic, because the "I think" which my apperception appends to each of my representations cannot ever be derived by mere analysis of each representation.
• Transcendental apperception is a priori, because a representation is nothing if it cannot be taken into a numerically identical consciousness. The very concept of a representation includes the fact that it is being represented to consciousness.
• Since transcendental apperception is synthetic a priori, it must have a priori rules. If transcendental apperception is a product of the pure understanding (and Kant assumes that it is), the categories, as the rules of the pure understanding, must serve as its rules.
• As such, the categories must necessarily be applied to all representations that are taken up into transcendental apperception (i.e. all of our representations).

The Transcendental Deduction (in the Second Edition, aka. the B-deduction)

The objective of the Transcendental Deduction is to remove the worry that the content of sensible intuition might not conform to the categories. In the second edition of the Critique, it is split into two parts. The first part concludes that the manifold in an intuition necessarily stands under the categories and, though this seems like a sufficient conculsion for the whole of the dudction, it is only the beginning; the second part argues that the categories are conditions of possibility for experience, and are therefore valid a priori of all objects of experience. -- The Transcendental Deduction therefore has two parts to make one proof.

For the first part of the Deduction, Kant argues that the categories are rules for the thought of sensible intuition's objects in general - that is to say, they function as rules for how the manifold of an intuition is determined with respect to the functions of judgment. As such, this first step abstracts from the spatiality and temporality of human sensible intuition, and focuses on the concept of an object of sensible intuition as such. Kant must therefore show that any representation which is brought into the unity of apperception is related to such an object (i.e. an object of sensible intuition as such) and standsion because it is the self-consciousness which produces under the categories. (Note: the 'unity of apperception' is the fundamental compilation of the entire manifold of our representations (which is a consciousness of multiple things), bringing them together in one consciousness. It is required for experience to be possible: imagine if ten people all were given one word of a ten-word sentence, but they couldn't share their words - no matter how much each person thought over their word, the sentence as a whole would never be thought. In this metaphor, each word is a representation and, if the people were to share their words to form a sentence, this would represent the unity of apperception.)

The Deduction requires its second part to show that the categories aren't just conditions of possibility for thought, but for experience, which is more than just discursive thought. To show that they are the conditions for experience, Kant must show that the categories relate to the forms of sensibility (namely space and time), thereby demonstrating that the categories are conditions of perception (as well as thought).

A Condensed Form of the Notes Above:

• In order for an object to be grasped as object, the manifold representations of that object must be combined.
• The combination (synthesis) of a manifold of representations doesn't come from an intuition or from the form of intuition; however, it does come from the subject, since nothing can be combined in an object which is not already combined in the subject. Synthesis of the manifold must therefore be a spontaneous act of the understanding.
• Synthesis of a manifold depends upon the unity of the manifold; whatever grounds the unity of the manifold therefore grounds synthesis, and therefore grounds knowledge of objects.
• The unity of apperception - that is, the numerical identity of the "I" which possesses each representation in a manifold - is what allows for the unification of representations. In other words, it grounds the synthesis of manifold representations.
• Just as the possibility of all sensible intuition is grounded on the fact that the manifold of sensibility stands under space and time (as showed in the Transcendental Aesthetic), the possibility of thinking (i.e. the use of the understanding) is that the manifold of all intuition stands under the unity of apperception.

Transcendental Analytic, Second Book: The Analytic of Principles

General logic deals with concepts, judgments, and inferences, which correspond to the "higher faculties of cognition": the understanding, the power of judgment, and reason.

Formal logic abstracts from all content of cognition, concerning itself only with the form of thinking. It can therefore include a canon for reason, analysing the actions of reason into their consitutent parts, regardless of what content these actions are concerned with.

Transcendental logic is concerned with content, of a particular kind: that of pure a priori cognitions. It cannot be divided in the way that general logic is, because the transcendental use of reason is invalid and doesn't belong to the "logic of truth" but to the transcendental dialectic.

The understanding and the power of judgment do each have a canon of objectively valid use in transcendental logic; the therefore belong to the transcendental analytic. It is only reason which is dialectical when it is put to transcendental content.

The analytic of principles is a canon for the power of judgment; it will show how this faculty applies the concepts of the understanding to appearances - in judgments.

Introduction: On the transcendental power of judgment in general.

While the understanding can be seen as the faculty concerning rules, the power of judgment can be seen as the faculty of subsuming under rules - determining whether or not something stands under a rule. Understanding provides abstract and general rules to supply the power of judgment with, but it is judgment alone that makes these rules concrete by applying them to things. -- When an example is given in an explanation of something, it strengthens the power of judgment by showing how particular rules are applied; conversely, it may go so far as to weaken the understanding, by showing a rule in a merely particular and not a universal setting.

Transcendental logic may be of benefit to the power of judgment, because the (a priori) concepts with which it is concerned are established a priori as objectively valid, and therefore come alongside a sufficient account of the conditions under which objects must fall in order to comply with them.

The Analytic of Principles is the transcendental doctrine of the power of judgment. It contains two chapters: (1) concerning the sensible condition under which pure concepts of the understanding can be employed; (2) concerning the synthetic a priori judgments (aka. the principles of pure understanding) which come from the pure concepts under these conditions and ground all other cognitions.

First Chapter: On the Schematism of the Pure Concepts of the Understanding

Whenever an object is subsumed under a concept, both the object and the concept must, as representations, contain the same aspects. In other words, they must be homogeneous. For example, a plate (object) has homogeneity with the circle (concept - of pure geometry) because roundness is thought in the former and intuited in the latter.

However, pure concepts of the understanding - as opposed to empirical intuitions - are never homogeneous. They can never be encountered in an intuition. This is because their content is purely logical (e.g. the category of substance is merely the category of something that is the subject of a logical predication), and the content of our experience isn't immediately presented to us in a logical structure. How, then, are empirical intuitions able to be subsumed under these categories? This question is why a transcendental doctrine of the power of judgment is necessary. The categories, unlike empirical concepts, have no resemblance in concreto to the objects which are subsumed under them. There must therefore be a mediating thing, which has homogeneity with both category and object, which makes possible the application of the former to the latter. This thing must be pure (i.e. without empirical content) and must be both intellectual and sensible. This mediating thing is a kind of representation, which Kant names the transcendental schema. The idea is that the a priori content of the categories will only be able to be applied to objects if it can be associated with an equally a priori (and universal) property that is immediately manifest in the experience that gives us these objects.

The concept of the understanding contains synthetic unity of the manifold in general. Time, which is the condition for the manifold of inner sense - and therefore is the condition of the connection of all representations - contains an a priori manifold in pure intuition. A "transcendental time-determination" is homogeneous with the category (which constitutes its unity) insofar as it is universal and rests on a rule a priori. Yet it is also homogeneous with an appearance insofar as time is involved in every empirical representation. As such, application of category to appearance is possible on the basis of transcendental time-determinations. In other words, every schema is a feature of the structure of time (or of relations in time), because this structure (a) shapes the experience in which objects are found and (b) is a priori and can therefore be associated with the categories.

So - the schemata are the a priori formal conditions of sensibility that determine how a category can be applied to an object. Each category has its own schema which restricts its use. The "schematism" is simply the procedure of the understanding in accordance with these schemata.

A schema is always a product of the imagination, but it is not an image. Three dots in a row - • • • - is an image of the number three. But a thought of the number three (or any other number) is the representation of a method for representing a particular multitude in an image, in accordance with a certain concept (e.g. "dot"). The thought isn't itself an image. This kind of representation of a method for the imagination is a schema.

There are four ways of thinking time; each way corresponds to a quarter of the table of categories.

• Time as a series grounds the schema of unity, plurality, and totality. Time as a series is generated by each moment in time being thought of as in a chain of succession (of seconds, hours, months, etc.). Time is therefore not just intuited, but thought of as a magnitude; and the schema of magnitude is number, which serves to capture the successive addition of one unit (a second, an hour, etc.) to the next. Thinking an object as existing in one such unit serves as the schema for unity; thinking an object as existing in a series of units is the schema for plurality; finally, thinking an object as existing in enough units to complete a greater series (e.g. seven days to complete a week) serves as the schema for totality. Overall, Kant actually says that number as such serves as the schema for all three of these categories.
• The content of time grounds the schemata of reality, negation, and limitation. The opposition of reality and negation is gronuded in the distinction of one particular time being either filled or empty. But the difference or transition between reality and negation/non-reality is made up of degrees (the difference is therefore thought as a magnitude) it is this division of the difference which grounds the schema of limitation.
• The ordering of objects in time grounds the schemata of substance, causality, and community/reciprocity. The persistence of something through time while other things change is the schema of substance; the schema of cause is the succession of particular states of a manifold insofar as this succession is subject to a rule; the schema of community/reciprocity is the coexistence of objects at the same time.
• The sum total of time in which objects exist grounds the schemata of possibility, actuality, and necessity: the schema of possibility is the agreement of a representation with the conditions of time in general - i.e. a representation occupying an undetermined time; the schema of actuality is the existence of a representation in a determined/particular time; the schema of necessity is the existence of a representation at all times.

Some examples, to clarify how this works: we can think of an object as actual if it is represented in a determined time; we can think of an object as a substance if it endures through a length of time; we can think of things as a unity when they exist in a single specific time.

Second Chapter: System of all principles of pure understanding.

Introduction

The schematism examined the conditions under which the power of judgment can use the categories for synthetic judgments. This second chapter will systematically enumerate the actual judgments that are brought about under these conditions. In effect, it will show what can be known a priori about possible experience. The relation these judgments have to possible experience constitutes all a priori cognition of the understanding. The relation they have to sensibility will systematically reveal the principles of the use of the understanding.

These judgments - the principles of pure understanding - ground all other judgments and are not themselves grounded in any more general judgments. But they are still provable: they receive a proof from the subjective sources of the possibility of cognition. (They couldn't receive an objective proof because they're not grounded in objective judgments).

The principles of focus in this chapter are those of the understanding (i.e. those that make use of the categories), not the principles of the Transcendental Aesthetic, which concern space and time (which themselves are restricted, not from schemata but from the fact that they can't be related to things in themselves). 'Mathematical principles' are also not relevant to this chapter, because they're derived from intuition and not the categories; however, the principles presented in this chapter are what provide the possibility for such mathematical principles.

The chapter starts by examining the principle of analytic judgments in order to distinguish them from the synthetic principles at focus.

Analytic judgments and their supreme principle.

The general, and negative, condition of all cognitions, regardless of their content, is that they don't contradict themselves. If they do, then they "are nothing". But a cognition, even if it isn't contradictory, can still combine concepts in an unjustified manner. Such a cognition is false/groundless, but not contradictory. The proposition that no cognition be contradictory (or, "that no predicate pertains to a thing that contradicts it") is the principle of contradiction. It is a criterion of all truth, since it doesn't concern the content of a cognition. It is also a negative criterion, but a positive use can still be made of it: if a judgment is analytic, its truth must be able to be cognized in accordance with the principle of contradiction. Because an analytic judgment states that a predicate does pertain to a concept that does indeed contain it. As such, the principle of contradiction is the supreme principle of all analytic cognition. For synthetic cognition however, it can only determine falsehood, and not truth.

Before proceeding to synthetic judgments, Kant points out how some (supposed) formulations of the principle of contradiction are, in fact, not analytic. For instance, "It is impossible for something to be and not to be" appears analytic, but is only actually true if it is appended with "... at the same time", thereby synthesising a predicate with the concept of time. A similar example is: "A person who is unlearned is not learned [at the same time]"; this, however, can be expressed analytically: "No unlearned person is learned" - this latter formulation is analytic because it doesn't imply any temporal relation.

The problem regarding synthetic judgments.

Explaining the possibility of synthetic - and then synthetic a priori - judgments, and thereby determining the boundaries of pure understanding, is the central task of a transcendental logic. We saw above that an analytic judgment is simply an assertion of identity (if it is affirmative) or contradiction (if it is negative); in a synthetic judgment, however, the relation between subject and predicate is neither one of identity nor contradiction; as such, the truth of a synthetic judgment cannot be found in the judgment itself. In order to make a synthetic judgment, and go beyond a given concept to compare it with another, then a third thing - which is separate from these two concepts - must be necessary.

Kant notes that there is only one totality in which all representations are contained: inner sense (and time, which is the form of inner sense). Furthermore, the schematism showed that the thinking of any representation is done by comprehending that representation in a certain determination of time. As such, the 'third thing' which is put alongside two concepts in a synthetic judgment is time.

Kant expresses this supreme principle of synthetic judgments as such: "every object stands under the necessary conditions for the synthetic unity of the manifold of intuition in a possible experience". This is the clue to discovering the system of the pure understanding's principles.

Principles of the pure understanding, "laws of nature", and empirical principles.

Laws of Nature are not examples of principles of the pure understanding; since they only apply to particular subsets of objects of experience, they are merely applications of more fundamental principles. -- These fundamental principles of the understanding contain the conditions for particular rules and laws, and experience provides the objects to stand under those rules and laws.

Principles of the pure understanding are easily distinguished from empirical principles, because only the former hold actual necessity. There are other principles (namely, those of a priori intuitions - space and time) which are necessary as well, but they won't be analysed here because they don't originate in the understanding.

The table of principles.

The principles are simply the "rules of the objective use of the categories"; as such, the table of categories provides a guide for how we should organise the totality of the principles:

Kant claims that the principles belonging to the categories of magnitude and quality (i.e. groups 1 and 2) are mathematical principles, and they concern individual intuitions (and the objects of these intuitions). The principles belonging to the categories of relation and modality (i.e. groups 3 and 4) are dynamical principles which concern the existence of appearances. They therefore require thought, and are of only "discursive certainty", whereas the former group are of "intuitive certainty".

Axioms of Intuition

The principles known as the axioms of intuition stand under the common principle: all intuitions are extensive magnitudes. Kant elaborates on this principle in several steps. He defines an extensive magnitude as one in which a representation of the parts is required for a representation of the whole. Space and time are the most immediate examples of extensive magnitudes, therefore, since they can both be broken down into smaller constituent parts (e.g. an hour into minutes into seconds). Now, given that all appearances are necessarily formed by the intuitions of space and time which they contain, all appearances also necessarily appear as extensive magnitudes.

Kant concludes that "it is is this [principle] alone that makes pure mathematics in its complete precision applicable to objects of experience … Empirical intuition is possible only through the pure intuition (of space and time); what geometry says about the latter is therefore undeniably valid of the former".

Anticipations of Perception

The common principle of all anticipations of perception is: in all appearances the real, which is an object of the sensation, has intensive magnitude. An intensive magnitude is a quantity which is not a totality of parts, though it can be expressed as a multiple of other quantities. For instance, while '5 metres' is extensive because it is the sum of 500 centimetres, the temperature '100 degrees Celsius' isn't the sum of 100 individual degrees - that is to say, it doesn't consist of 100 parts.

Kant's argument for this claim is quite simple: all perception contains sensation, i.e. that part of a perception which represents the "real". And "since sensation in itself is not an objective representation, and in it neither the intuition of space nor that of time is to be encountered, it has, to be sure, no extensive magnitude, but yet it still has a magnitude … an intensive magnitude'.

For some examples: the visual aspect of any sensation includes colour, which is in degrees (and is therefore an intensive magnitude); any aural sensation includes volume, which is also an intensive magnitude, and so on. All sensation includes some kind of intensive magnitude. As such, the mathematics that deals with degrees is necessarily applicable to experience.

Analogies of Experience.

The principle of the analogies of experience is that experience is possible only through the representation of a necessary connection of perceptions. Experience never consists of a completely individual perception; rather, it is a synthesis of a manifold of perceptions. This synthesis is not contained in perception – it is not the perceived objects themselves that ‘relate’ to one another inherently – but is carried out by the perceiver’s consciousness, through the synthetic unity of the manifold. Since the state of affairs presented in any experience is contingent, the necessity of the synthesis that forms the experience must originate in the perceiver and not in what is perceived. The synthesis can only be achieved in the first place if the objects of perception are perceived as coincident in time and space, and since time and space themselves cannot be perceived, the spatiotemporal coincidence of the objects must originate in their mental representation.