a priori, universality and necessity, all possible worlds, existence. The why......
“.....Our knowledge springs from two main sources in the mind, first of which is the faculty or power of receiving representations; the second is the power of cognizing by means of these representations. Through the first an object is given to us; through the second, it is thought....”
“....We apply the term sensibility to the receptivity of the mind for impressions, we call the faculty of spontaneously producing representations, or the spontaneity of cognition, understanding....”
“....Neither of these faculties has a preference over the other. Without the sensuous faculty no object would be given to us, and without the understanding no object would be thought. Thoughts without content are void; intuitions without conceptions, blind. Hence it is as necessary for the mind to make its conceptions sensuous (that is, to join to them the object in intuition), as to make its intuitions intelligible (that is, to bring them under conceptions). Neither of these faculties can exchange its proper function. Understanding cannot intuit, and the sensuous faculty cannot think. In no other way than from the united operation of both, can knowledge arise....”
“....We therefore distinguish the science of the laws of sensibility, that is, aesthetic, from the science of the laws of the understanding, that is, logic....”
“....logic in its turn may be considered as twofold—namely, as logic of the general use, or of the particular use, of the understanding. The first contains the absolutely necessary laws of thought, without which no use whatsoever of the understanding is possible, and gives laws therefore to the understanding, without regard to the difference of objects on which it may be employed....”
“....general logic has nothing to do with the origin of our cognitions, but contemplates our representations, be they given primitively a priori in ourselves, or be they only of empirical origin, solely according to the laws which the understanding observes in employing them in the process of thought, in relation to each other. Consequently, general logic treats of the form of the understanding only, which can be applied to representations, from whatever source they may have arisen....”
“....All intuitions, as sensuous, depend on affections; conceptions, upon functions. Conceptions, then, are based on the spontaneity of thought, as sensuous intuitions are on the receptivity of impressions. Now, the understanding cannot make any other use of these conceptions than to judge by means of them...”
“....In all judgements wherein the relation of a subject to the predicate is cogitated, this relation is possible in two different ways. Either the predicate B belongs to the subject A, as somewhat which is contained (though covertly) in the conception A; or the predicate B lies completely out of the conception A, although it stands in connection with it. In the first instance, I term the judgement analytical, in the second, synthetical...”
“....The former may be called explicative, the latter augmentative judgements; because the former add in the predicate nothing to the conception of the subject, the latter add to our conceptions of the subject a predicate which was not contained in it, and which no analysis could ever have discovered therein....”
(Insert for clarity: “...analyse the conception, that is, become conscious of the manifold properties which I think in that conception,...”)
“....Judgements of experience, as such, are always synthetical. For it would be absurd to think of grounding an analytical judgement on experience, because in forming such a judgement I need not go out of the sphere of my conceptions, and therefore recourse to the testimony of experience is quite unnecessary....”
And the wherefore.......
“....But as in all the attempts hitherto made to answer the questions which reason is prompted by its very nature to propose to itself, for example, whether the world had a beginning, or has existed from eternity, it has always met with unavoidable contradictions (...), but it must be possible to arrive at certainty in regard to the question whether we know or do not know the things of which metaphysics treats....”
“....For what of analysis, that is, mere dissection of conceptions, is contained in one or other, is not the aim of, but only a preparation for metaphysics proper, which has for its object the extension, by means of synthesis, of our a priori knowledge. And for this purpose, mere analysis is of course useless, because it only shows what is contained in these conceptions, but not how we arrive, a priori, at them; and this it is her duty to show, in order to be able afterwards to determine their valid use in regard to all objects of experience, to all knowledge in general....”
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Logic is the law of thought, understanding is the means of thought, therefore logic rules understanding. Understanding manifests in judgements, judgements are of two kinds, one for objects, one for thought of objects. Therefore the laws of thought in the form of logic rules judgements. One form of logic, in which thought is transposed to intelligible communication, is the proposition. A logical proposition requires a subject and a predicate, and the relationship between them determines the kind and the source of the judgement being communicated.
Analytic propositions dissect conceptions in order to determine whether the conception in the predicate belongs to the conception in the subject by the logical law of identity, re: all dogs are canines.
Propositions where the conception in the predicate does not belong to the conception in the subject by identity, but must be connected to it by synthesis from an intuition of experience, earn the title synthetic propositions, re: all dogs have four legs.
Analytic judgements a priori occur when the relations in the conceptions in the predicate belong to the conceptions in the subject from the logical principles of universality and necessity, re: all bodies are in space.
Synthetic a priori judgements, synthetic in the method of their subject/predicate relation and a priori from the nature of the conceptions in them, are shown by any pure mathematical expression.
An analytic a posteriori proposition may be merely a tautology, hence useless, re: my hat is a hat.
Synthetic a posteriori propositions are redundancy in kind, for synthetic propositions are already completely empirical judgements, the concept in the predicate supplements the concept in the subject, even if not contained in it.
The proof of the possibility and validity of synthetic a priori propositions is given in mathematics and the principles for them are given to all natural sciences. Their true value, however, lays in their employment by reason in questions of metaphysics, which is in effect, the examination of grounds for possible truth.
Thus is the division into and function of differences in the approach to the attainment of knowledge, and the justification for synthetic a priori judgements being by far the most important.
Theoretically.