Symbolize "Absence of evidence is not evidence of absence"

ektrules

This isn't homework. I was bored and ran across a blog that claimed the statement, "Absence of evidence is not evidence of absence" wasn't logical. It seems like common sense to me, so I was going to try prove it to be valid with symbolic predicate logic (again, I was bored), but I quickly became confused on how to correctly symbolize it in a way that I could prove it.

So, how would one symbolize this statement?

Hurkyl

It's really an issue more akin to statistical inference than formal logic. I suppose the closest analogy to formal logic would be the fact the following is not true:

In a formal theory, if P is indeterminate, then "not P" is a theorem.

EnumaElish

"( (Premise ==> Conclusion) and (not Premise) ) ==> not Conclusion" is false ???

ektrules

Well, the definition of a valid argument is if all the premises are true, then the conclusion must be true. So, a premise could be false, and the conclusion could be true, and it could be a valid argument. So yeah, "( (Premise ==> Conclusion) and (not Premise) ) ==> not Conclusion" is false.

Although, I think "( (Premise ==> Conclusion) and (not Premise) ) ==> not Conclusion" is not an accurate symbolization of the original sentence (though it may be equivalent). I was trying to symbolize the sentence to produce a formal proof, and can't get my head around how go do it. Perhaps second-order logic is needed (I don't know anything about that)?

EnumaElish

Yes; I was suggesting

"..." is false

as a logical formulation of "absence of evidence is not evidence of absence," although somewhat more generally (any premise as opposed to your specific premise, "evidence"), and hence the question marks at the end.

techmologist

As Hurkyl said, the problem refers to statistical inference, so you will need more than logic to prove or disprove the statement. You need some formal account of what it means for some event to be evidence (confirmation) for some hypothesis.

In the Bayesian theory of confirmation, absence of evidence is evidence of absence. The fact that this is not in line with the common sense notion of evidence is a possible argument against the Bayesian approach, but it also has a lot going for it. In this theory, an event e confirms hypothesis h if P(h|e) > P(h). That is, the conditional probability that the hypothesis is true given the occurrence of event e is higher than the prior probability that the hypothesis is true.

My interpretation of the statement "absence of evidence is evidence of absence" in symbolic terms would be

P(h|e) > P(h) implies P(~h|~e) > P(~h)

which happens to be true. But note that this refers to a particular piece of evidence, e. The above implication just says that if the occurrence of some event confirms a hypothesis, then the non-occurrence of that event disconfirms it. I'm not sure how to deal with the statement if it is interpreted to mean "complete absence of any kind of evidence."

EDIT:
Also note that absence of evidence is not necessarily good evidence for absence. If the prior probability P(e) is very small, then the non-occurrence of e may only disconfirm hypothesis h very slightly. This might mean that the discrepancy between Bayesian evidence and common sense evidence is not so big after all.

Hurkyl

If we have observed neither "e" nor "not e", then we can infer nothing, and are merely speaking hypothetically.

EnumaElish

How would one formalize the statement 'we have observed neither "e" nor "not e" '?

My instinct is to say "there is a large error around e, so I don't have a good sense whether what I am observing is an event, or just random noise."

techmologist

Yeah, that's true. Propositions about the future don't have a truth value. It's funny, I wasn't even thinking about formal "laboratory-style" experiments where the result comes unambiguously after some procedure. And those are the prototypical scientific experiments. I had in mind more informal experiments that are sort of continually taking place. Like, if such-and-such thing existed/happened, we would expect to have seen some evidence of it by now.

So what I really meant to say was "the complete absence of any confirming evidence", assuming that there had been time for some kind of evidence to surface. Think Russell's tea pot.

EDIT:

Or not. I guess the whole point of Russell's tea pot is that there is no opportunity for it to make itself evident.

Tac-Tics

I think it depends on its interpretation. Does "absence of evidence" mean evidence has not (yet) been witnessed? Or does it mean that evidence cannot (ever) be produced?

The traditional meaning of the phrase (in reference to god and religion) is the former. However, the latter is more pertinent to symbolic logic.

Jerbearrrrrr

Evidence isn't really related to logic.

We don't have any kind of valid 'induction' to extend facts about things that are here to facts about things that aren't here.

There might be a region in space where gravity is repulsive, and all magnets are monopoles.

Because evidence can't prove anything, it's sort of independent of the actual truth value of a statement. I think "absence of evidence" and "evidence of absence" are things that don't actually affect the rest of the universe a priori.

Tac-Tics

It is entirely related.

In logic, we sometimes use the word "evidence" synonymously with "proof". A formal proof is all the evidence you need for something to be true in logic.

Certainly though, you can't prove something like infinity of primes by counting on your hands and fingers. You can provide evidence (proof) that this number is prime or that number is prime, but collecting even a billion such cases doesn't do any good towards providing evidence of the infinitude.

EnumaElish

http://www.bmj.com/cgi/content/full/311/7003/485

qemist

Never seemed right to me. When I look in the cookie jar and see no evidence of cookies apart from a few crumbs, that's pretty good evidence that cookies are absent from the jar.

Hurkyl

Evidence of absence counts as evidence of absence.

DaveC426913

The point is that you have artificially restricted your scope to the cookie jar.

What if you were on an open football field? Now what can you say if you don't see cookies?

Tokipin

Could the likes of Doxastic logic be usable here? I'm not really familiar with formal logic, just throwing this out.