I believe that Kant defended the "law of cause and effect" by stating this argument: (P) If we didn't understand or acknowledge the law of cause and effect, we couldn't have any knowledge. (Q) We have knowledge. Therefore: (P) we acknowledge the law of cause and effect. Isn't this line of reasoning a fallacy? P implies Q, Q, : P

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It seems to me you haven't reported the inference accurately. The conclusion, "We acknowledge the law of cause and effect," is the negation of the antecedent of (P) and not, as you report, (P). (That is, your premise (P) is of the form: if not-X, then not-Y. And the conclusion of your argument is X.) So, the argument really has the form "If not-X, then not-Y" and "Y", therefore "X". This is a correct form of inference in classical logic.

You're right that "If X, then Y" and "Y" do not imply "X"; that is indeed a fallacy. But this argument is rather of the form: "If X, then Y" and "not-Y", therefore "not-X". And that is a correct inference.

You have certainly put your finger on a complex issue. One might say you've got a dragon by the tail.

First, I should call your attention to the fact that you've rendered his argument in two logically different ways. The first rendering is actually a valid form of deductive inference, not a fallacy. Philosophers, in their pretentious way, call it a modus tollens. The terms in which you've put it allow for this rendering:

1. If Not-P, then Not-Q.

2. Q.

3. Therefore, P.

And, by the way, that first rendering can also be restated in another valid form called a modus ponens:

1. If we have knowledge (Q), then we understand or acknowledge the law of cause and effect (P).

2. We have knowledge (Q).

3. Therefore, we understand or acknowledge the law of cause and effect (P).

There's a rather large issue lurking here, too, as to what "understanding" and "acknowledging" mean, how they're similar, how they're different. (See, for example, Stanley Cavell's, "Knowing and Acknowledging" in his book, Must We Mean What We Say? , Cambridge UP, 1976). But let's put that aside for the moment.

The second rendering you present (the way you put it in your very last sentence) is, take note, different from the first; and it is indeed fallacious. It's a fallacy known as affirming the consequent. That second rendering might be restated this way:

1. If we understand or acknowledge the law of cause and effect (P), then we have knowledge (Q).

2. We have knowledge (Q).

3. Therefore, we understand or acknowledge the law of cause and effect.

Okay, enough of the logic of your question. More importantly, I'd like to observe that while Kant's transcendental "deduction" of the idea that general causal judgments are true in a what philosophers call an "a priori" way might be called an argument, it's an argument of a rather unusual sort. I would caution you therefore about the risks of rendering Kant's argument in the simple deductive way you present it here. I think this sort of rendering masks the special way he thinks he is "proving" his case. Kant himself remarks that his proof is more akin to the judgment of a kind of legal tribunal than a proof of deductive reasoning. At the risk of butchering Kant too much, I'd suggest you consider Kant's "deduction" as a bit closer to this rendering:

1. A necessary condition for the possibility of knowledge of the natural world is the a priori truth of the claim that every event in the natural world has a natural cause. (In short, scientific knowledge is possible only if the law of cause and effect is a priori true.)

2. Let's proceed as if scientific knowledge is possible.

3. It's an a priori truth that every event in nature is caused by another event in nature (i.e. that the causal law is true).

Of course, that's a rough rendering, and I hope my colleagues will forgive me for its brutality; but note how my second premise is much more guarded than yours, and note how I emphasize that Kant is more interested in figuring out what the necessary conditions for the possibility of knowledge are than proving that we actually have knowledge. Anyway, I hope this helps you pursue a more advanced understanding of the issue.

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