Men of Straw, Invisible Black Sheep and the Tale of Two Lemmas
The Straw Man Fallacy:
Detailing the logical errors in most of their arguments would require a book, but there is one logical fallacy that the "alties" seems particularly fond of: the "straw man." For those who didn't take Professor Petersen's Logic class, the straw man fallacy is simply attacking an argument that your opponent didn't make. Usually, this is done by implying (or asserting) that your opponent has argued something easy to refute - usually something extreme and indefensible - and then attacking that argument instead of the one your opponent actually made.
As a hypothetic example, if your opponent said that there is no data supporting a causal connection between autism and mercury, you could attack them for saying that mercury is completely safe. You could ask them, "Do you think that mercury is a good thing to inject into children?"
This tactic does not refute your opponent's argument - it doesn't even address it - but it is a good way to switch the discussion from an argument you can't win (since you don't have any decent data) to one that you can win. It's not intellectually honest, but it works more often than not. Just watch any political campaign.
Some of the more popular straw man arguments that I've seen used in "alternative" medicine are:
 "People should be allowed to make medical decisions for themselves."
This straw man is usually used to dodge the issue of practitioners failing to be honest about the data supporting their treatments (usually none). By making it appear to be an issue of personal autonomy, rather than an issue of truth in advertising, the builders of this straw man hope to avoid the tough questions about a practitioner's obligation to be honest and open about what they are selling.
 "You're trying to take away our right to free speech!"
The First Amendment of the US Constitution, and similar constitutional guarantees in other countries, does not grant carte blanche rights to say anything one wants. Libel, slander, and mail fraud are all offenses that involve the improper use of speech. The right to freedom of speech and press is not a license to lie, deceive and defraud.
Argument from Ignorance:
Another favorite "altie" fallacy is usually expressed as, "You don't have any proof that it doesn't work (or cause cancer/autism/terminal moraine)."
This fallacy - the argument from ignorance - is used in much the same way as a straw man. It is much easier to argue that something isn't proven to never happen than to argue that it does. A classic story of this (told to me by the aforementioned Prof. Petersen) is the story of the two logicians:
On a train through Scotland, a senior logician and his junior colleague were observing the countryside. Upon sighting a huge flock of grazing sheep, the junior colleague casually remarked, "There's not a single black sheep in that whole flock."
The senior logician, roused from his doze, retorted, "You can't say that - you don't know if you can see the whole flock!" Chastened, the junior logician replied, "Well, all the sheep in view are white." The senior logician snorted into his mustache and said, "You can't say that, either!" "Well!" said the exasperated junior member, "What can I say?" With a smug look, the senior logician replied, "That all the sheep in view are white, on at least one side!"
So, the absence of visible black sheep does not eliminate the possibility of black sheep - no matter how many sheep are observed - because the presence of a single black sheep somewhere in the Universe (Past, Present or Future) firmly establishes the existence of black sheep.
However, in biology, it is not usually important to know if black sheep exist, as long as you know that they are exceedingly rare. For example, while it has not been possible to definitively prove that power lines do not cause brain cancer in children, it has been shown that the risk is between zero and a risk so small that it blends into zero (i.e. the uncertainty range includes zero).
Likewise, it may not be possible to prove that Dr. Alt's Sure-Fire Cancer Cure doesn't ever cure cancer, but it is possible to show that it happens less often than with other, more established treatments (or placebo, or the spontaneous remission rate of the cancer). So, even if it is true that someone (or even two or three someones) had their cancer "disappear" after taking Dr. Alt's treatment, it doesn't mean that it actually works.
One logical fallacy that shows itself time and again in "alternative" medicine is the false dilemma. This fallacy is basically a failure to consider (or present) all of the possible alternative choices. In "alternative" medicine, this fallacy is often somewhat hidden, as in the following:
"If we let the government regulate vitamins and supplements, then you won't be able to buy them anymore."
In this case, the false dilemma presented is between the following two options:
 Unregulated advertisement and sale of vitamins and "supplements".
 Vitamins and supplements being banned from sale.
In reality, there are many other possible choices, such as:
 Vitamin and supplement retailers being required to provide sufficient data to support any claims they make in advertising or on the label.
 Limiting sales of vitamins and supplements to only those items whose safety has been established by legitimate studies.
 Requiring that vitamin and supplement makers (and sellers) make no health-related claims about their products.
 ... etc.
By using a false dilemma - an "either this or that" type of framework for the debate - the "alties" manage to make their preferred stand at least the lesser of two evils. However, they can do so only by picking an extreme option and making it appear to be the only other choice. If all the potential options are placed on the table, their preferred choice is the one that looks extreme.
Well, that's all for now. Class dismissed!