Kant's Second Analogy is where he takes his stand against Hume's skepticism about causality without lapsing into Leibnizian dogmatism. Its aim is to show that every change is subject to causal law. The "Proof" is lengthy and repetitious, so I have isolated one paragraph in which something approaching a self-contained argument is given. This paragraph was added in the second edition, so it represents Kant's considered judgment on the matter.
"All alterations take place in conformity with the connection of the law of the connection of cause and effect." Alterations are changes in the states of a substance which remains through the change. At one time the substance is "determined" as being in state A and at a subsequent time it is "determined" as in state B, which is opposed to A. Thus there is a succession of states, one occurring first and the other subsequently. This succession is "an objective relation of appearances that follow upon one another."
The term "determined" is crucial to the argument. It means that an individual subject cognizes the substance as being in the successive states. A cause is a state which necessarily determines another state as its consequence. Thus the claim is that whenever an alteration takes place, the subsequent state B is necessarily determined by the antecedent state A.
The argument invokes a number of human cognitive faculties. Perception is simply representation with consciousness. It has no pretense to objectivity: intuition is an objective perception. To say that "I perceive that appearances follow one another," then, is simply to say that there is a representation of two different states, one coming before the other, but this is not to say there is any succession in the represented object. Moreover, the "perception" of succession is not passive, but rather is a connection, a product of the faculty of imagination, which "sets one state before and the other after."
The Argument
The key premises of the argument are these:
1) If an alteration has taken place, one state has succeeded another in the object, in a definite order.
2) If one state has preceded another in the object, the objective succession can be determined only by means of a rule according to which the prior state must occur before the subsequent state.
3) A rule according to which a prior state must occur before a subsequent state is a causal law.
4) Therefore, if an alteration has taken place, the fact that it has occurred can be determined only by means of a causal law.
5) The substances which are altered are appearances, and the means of their determination is what makes them objects of experience.
6) Therefore, if an alteration in an object of experience has taken place, it is subject to a causal law.
Premise 1) is said to be settled by the First Analogy. Premise 2) is defended on the grounds that mere perception cannot determine the objective order of succession, since the imagination can place perceptions in any order, and there is no perception of time itself to serve as a standard of perception. Premise 3) is definitional. Premise 5) is an expression of Kant's transcendental idealism. For the fact that we can determine succession only with respect to a causal law is relevant to an object's standing under a causal law only if the determination makes the object what it is.
It would take some more work to make the argument formally valid, but let us concede that it is. Premises 2) and 5) are the most obviously controversial. The fact that I can imagine things in a different order than that in which they occur is of no consequence so long as I am able to tell generally when the order is merely the work of the imagination. If it be claimed that there is a transcendental faculty of imagination which is responsible for all the perceptions I have, it still is not clear that it can operate arbitrarily to give the illusion of succession at any point. As regards transcendental idealism, it is not a very palatable doctrine in its own right, and Kant's arguments for it are weak.
I believe there is a stronger argument beneath the surface. Descartes set the problem of distinguishing the objective in general from the subjective, and the standard answer is that we can know that our perceptions are objective because of the coherence of the objective. It could be claimed that coherence depends on laws of nature, which serve as standards against which any given perception is measured. If a person perceives a sequence that is in conflict with laws of nature, the perception is judged to be subjective. Thus in order to be able to distinguish objective perceptions from subjective perceptions in general, laws of nature are needed.
Kant can be seen as trying to exploit an assumption Hume made uncritically: that we experience alterations as taking place. He never questioned our ability to ascertain whether B succeeds A, but only whether, given A, B must occur. Kant's argument attempts to establish that the necessity of succession is a prerequisite for there being an alteration at all. If this is so, then the grounds of Hume's skepticism, that can never find in A the condition for B, are removed from under him. It does not matter that an examination of A shows no traces of B. What matters is that if the succession from A to B is to be considered as objective, some causal law must be operative.