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Particular type of "argument from ignorance" is "absence of evidence is evidence of absence."

I have proof that it's in fact true.

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Does it mean that argument from ignorance can be right?

If not - why? And if yes, then how do we know when it's fallacious and when it's right?

P.S. Replaced "valid" with "non-fallacious" in the title.

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    A sequence of formulas does not mean or prove anything. I am guessing this refers to Bayesian updating of probabilities based on evidence, but you'll have to spell out how you think this applies to arguing from ignorance. In some contexts, arguing from ignorance can be situationally valid (this is true of most informal fallacies), see Absence of evidence is not evidence of absence, but at present it is unclear what you have in mind, or what you are asking. – Conifold Jun 20 '19 at 19:17
  • @Conifold "A sequence of formulas does not mean or prove anything." Only if you don't know what they mean. "how you think this applies to arguing from ignorance." We have a test where instead of negative result we get absence of positive result (i.e. we performed test and failed to get positive result). We can then use failrue of getting positive test result as evidence for counter-hypothesis ~B. I.e. Bayes theorem will make us more sure about ~B being true. – user161005 Jun 21 '19 at 1:46
  • I do not know what they mean to you, in particular, what the argument based on them is supposed to be. That is why I asked to add it to the post. The best I can say now is that if there are prior reasons to expect detection of A when it is present, and it is not detected, then it would be evidence of its absence. That is a typical context for Bayesian updating of probabilities. But that is not an argument from absence of evidence alone. – Conifold Jun 21 '19 at 8:49
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I think at least in your titular question, you're mangling terms in a way that is unhelpful.

In logic and critical thinking, valid refers to when a deductive argument is such that if all of its premises are true, then its conclusion must be true (there are other similar but more exotic formulations -- e.g. must be constructable as a model -- but those don't change the point here).

Fallacy is term with two meanings that are both "errors of reasoning." Deductive fallacies are known forms where an argument is presented as deductive but is not valid. For instance,

"affirming the consequent" = If A, then B. B. Therefore, A.

Such errors are damning to any argument that presents itself in that way.

"fallacy" has a second meaning which refers to bad reasoning in an informal sense. "appeal to ignorance" is an informal fallacy of this short.

"Fallacy" as used in the latter sense doesn't automatically decide whether someone has committed an error that destroys the argument. Instead, it often devolves into an argument about whether the claim in question is fallacious to use in that context -- since it is not definitely an error in formal reasoning.

For instance, "ad hominem" is an informal fallacy, but "you shouldn't trust him, because he's lied to you 100s of times" is not guilty of an "ad hominen" attack on his character, it's good reasoning as to why you should not trust him. (whereas "you shouldn't trust him, because he has brown eyes" is fallacious and ad hominem).

Given this, it's unsurprising that one can find an argument which appears to use the fallacy and prove something correctly. This doesn't disprove that it's possibility to fallaciously confuse absence of evidence with evidence of absence. It merely proves that the fallacy is informal. And in all likelihood proves that when applied to a controversial case that people will find grounds for disagreement.

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  • In other words, "It's a rule of thumb and what you presented is just particular case when this rule of thumb doesn't work". Did I get your idea correctly? – user161005 Jun 21 '19 at 9:07
  • That's close to what I'm saying -- but perhaps worded differently: informal fallacies rarely settle arguments, because we can argue for years about whether a usage is fallacious. – virmaior Jun 22 '19 at 0:37
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After thinking about it I think the answer is following:

Yes, it can be the case that argument from ignorance can be non-fallacious, given that we define it as using absence of evidence, rather than absense of proof and that the resulting argument is probabilistic, rather than deductive (i.e. it doesn't make us either 100% or 0% sure). But not always, it's valid for situation that was described in the above proof. In the proof we have absence of evidence that is at the same time the negative test result. We made successfull attempt to get test result (i.e. we successfully performed a test), but we didn't get the positive test result. In these circumstanses absence of the positive test result is the same thing as getting the negative test result.

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