I’m not sure of the argument from ignorance and of the concept of probability.

These are good questions. Keep 'em coming!

How do we know that it doesn’t follow?! Because of the lack of evidence for a causal relationship between A and B? If so, then isn’t this an argument from ignorance itself? I feel like something is wrong here.

Basically, both forms of the fallacy rely on the fact that there's no evidence in a certain way (no evidence A is false, or no evidence B is true). The thing is, you can only find evidence if you've gone looking for it. If no one has done that, then it's tenuous to make claims about the evidence we have on hand. This is also known as negative proof. So if I say, "my car is in the garage" for example, I can't assert the claim merely based on the fact that there is no evidence it isn't true - no one has bothered check the garage in the first place!

That said, there is something called negative evidence. This is disconfirmation of a claim or theory. If someone looked in the garage and didn't see a car, then this would be negative evidence against my claim (and actually prove it false). Negative evidence isn't a fallacy, because the claim was actually empirically investigated, which is how the negative evidence was found.

Moreover, if lack of evidence does not  at least  provide rational basis for disbelief in something, then really just tell me what’s the point in denying the existence of fairies, dragons, ancient aliens, demons, angels, or really anything mythical? How come concluding that these characters  at least most likely do not exist because of lack of evidence count as an “argument from ignorance” and thus fallacious?

Lack of evidence doesn't necessarily mean you can draw any conclusions about the existence of the thing you're discussing. But sometimes, evidence fails to show up where you would expect it to (see my 'car in the garage' example - if the car were there, you'd expect it...to be there). In those cases, since there's no plausible mechanism to explain the lack of evidence, it is rational to take a position of disbelief. Strictly speaking in a probabilistic sense, you might not say "they don't exist", rather, "it is extremely unlikely that they exist" (would be awesome if dragons were real, though - I'd love to ride one!)

But, if lack of evidence provides at least a rational basis for disbelief in something, then there’s another problem! Let’s say a supernaturalist presents a phenomenon that has no known natural explanation as of now. They then conclude it’s rational to reject there being a natural explanation because of the lack of evidence for there being one, and so it’s reasonable to accept the supernatural explanations. Again, isn’t there something wrong here?

This would fall under confusing currently unexplained with unexplainable. There is no known natural explanation, but that does not mean there will never be one. It depends on what new evidence comes in and what it suggests. Indeed, many things we once attributed to the will of God (like recovery from disease) now have naturalistic explanations (modern medicine, your immune system, etc.)

Furthermore, what's the base likelihood the supernatural explanation is true? Where is the evidence for it? Again, if they're relying solely on the lack of evidence against the proposition, that'd be an argument from ignorance. This is especially important because supernatural claims are often unfalsifiable - i.e. they're written in a way that makes them 'immune' to all disconfirming evidence.

I then make 100000 arguments like this seemingly terrible one. If I then calculate the total probability, it would be much higher, like perhaps 10%! I could keep making these terrible inductive and probability based arguments to reason any magical character into existence!

I know it's just an example, but your numbers would be pretty off. What you seem to be saying is that, if you make enough nonsensical inductive arguments, the total probability of one of them being true would be ~100% - but this is impossible, given the exclusion of other theories that have far more evidence behind them, like the Big Bang or even Steady State. Since the 'it's magic' explanation would fail to explain a lot of modern scientific observations, its base probability of being true would likely be infinitesimal, so even if you stacked lots of them...it wouldn't make much of a dent.

Secondly, each one of those arguments might conclude that a different magical character was responsible for the world's creation. E.g. if I say, "either God or Allah made the world". Is the probability this is true - P(G u A) - greater than P(G) or P(A)? Probably. But it doesn't increase the probability that any one of them is true, since they're different characters.

I hope that helped - my explanations might have been a bit confusing, and admittedly, I struggled to phrase the answer myself; a good way of challenging my views, of course.

Hi, Alex!

         Knowing that a conclusion does not follow is the same thing as knowing that the argument for it is invalid: that there is a possible circumstance in which the premises are true and the conclusion false. There are several different methods used to prove that an invalid argument is invalid. None of these methods commits the arguments from ignorance fallacy because the methods work by showing that there is a situation in which the premises of the argument are true and the conclusion false, which is enough to show that the argument is invalid.

How come concluding that these characters… Please please please explain to me this point thoroughly.

         People, at least those who are reasonable, do not disbelieve in dragons, ancient aliens, demons and angels merely because of a lack of evidence for them. Rather, generally they disbelieve because of lack of evidence in conjunction with the unstated assumption that there would be evidence for these beings if these beings did exist. Take God, for instance. An atheist might disbelieve in God because there is no evidence for him but that if he did exist , then there would be evidence for him. (For example, some might think that if there were a God, we should see a world with much less evil.) This assumption about the evidence that would be present if the beings in question existed makes the argument for disbelief valid. The assumption is so implicit that it is often rather forgotten, casting the illusion of an argument from ignorance. Disbelief is not grounded in an argument from ignorance, but rather in an argument whose key premise is implicit. That is my interpretation of the situation, at least.

Let’s say a supernaturalist…

         If the absence of a known naturalistic explanation for A is reason to accept a supernatural explanation, then by that same logic the absence of a known supernatural explanation of A is reason to accept a naturalistic explanation. Now, suppose there is neither a known naturalistic explanation nor a known supernatural explanation of a phenomenon. The supernaturalist’s way of thinking (absence of evidence for x is reason to reject x) would have us reject both explanations, which is absurd in itself and also contradicts the supernaturalist’s claim that we should accept the supernatural explanation.

         At the very least, however, the supernaturalist's reasoning cannot alone decide to reject one explanation instead of the other. For if there is neither a known naturalistic explanation nor a known supernaturalistic explanation, then his way of thinking equally supports the rejection of either explanation. The point is that if the supernaturalist cannot support his position outside of just saying that there is no known naturalistic explanation, then that lack of positive knowledge of the supernatural explanation would just be, by his reasoning, confirmation of the alternative explanation. In other words, the reasoning process behind the argument from ignorance is, for the person reasoning in this way, a double-edged sword. I may answer the rest of your post, as time allows.

Thank you, Alex

From, Kaiden