The Analogies of Experience are supposed to provide principles for the determination of objects with respect to time. The most general aspects of time with respect to the way objects of experience can be represented are: duration, succession and coexistence. The First Analogy concerned the way in which we can consider objects as enduring, and that is as permanent substances with transitory accidents. Kant claimed that all change is the replacement of one accident of an existing substance with another, such that creation or annihilation of substance is impossible. The justification of this principle is based on the claim that without it, the experience would lose its unity, specifically because it would allow the possibility of two distinct time streams.
It is essential to Kant's argument that the objects with which he is concerned are those of experience, and not things in themselves. Everything turns on the claim that without the "Principle of Permanence of Substance," there would be no unity of experience. Thus the principle is a necessary condition for the determination of appearances, which is their representation with respect to the time-relation of duration. Kant remarked that the principle does not apply to things considered in themselves, or as obvjects of the mere understanding. "Despite their being substances they cna be regarded, in respect of their existence, as depending on a foreign cause. But our terms would then carry with them quite other meanings, and would not apply to appearances as possible objects of experience" (A206/B252).
The Second Analogy provides a principle by which we determine objects of experience with respect to the time-relation of succession. The "Principle of Succession in Time, in accordance with the Law of Causality" states that any change in time must conform to causal law. All appearances are subject to laws, according to which their present state is a consequence of some prior state of appearances which are sufficient for the present state's existence. For a discussion of the terminology of the Second Analogy, see the Lexicon item Cause and Effect.
Kant was trying to provide a justification for the application of the principle of sufficient reason to objects of experience. He believed that no one had ever provided any kind of justification for the principle. Leibniz and his followers had failed because they considered the principle in its full generality, as applying to things in themselves. Wolff in particular had tried to show that it follows from the principle of contradiction, and Kant believed that the proof must fail, with the consequence that the principle is synthetic rather than analytic.
Hume had accepted the synthetic character of the principle and proved that it cannot be justified a posteriori. Most notably, Hume recognized that through experience we can never detect the necessary connection which is implicit in the concept of a cause. (For an exposition of Hume's views, see the entry on Hume as one of Kant's predecessors.) The only remaining avenue to justification of this synthetic principle, Kant concluded, is a priori. As with the category of substance, that of causality is required for the unity of experience.
The argument begins with the concept of an alteration, a change in the state of a substance. The determination of an alteration in time requires that one state be thought as existing before the change and the other one after. Thus the states are successive, occurring in a definite time-order. But the order of time itself cannot be perceived, since time itself cannot be perceived. The only evidence we have for the order of succession in appearances (the objective succession) is the order of succession in our perceptions (the subjective succession). But perceptions can occur in an order different from an order in the objects perceived. Kant held that the order of perceptions is a product of the imagination, and that the production of images may take place arbitrarily. To be able to tell when a succession takes place in the objects, that is, when an objective change has occurred, we must appeal to rules according to which the succession must take place in a certain way. Thus for the determination of succession in objects of appearances, we require the applicability of the principle of sufficient reason, which was to be proved.