Symbolize Absence of evidence is not evidence of absence

[ektrules]

Symbolize "Absence of evidence is not evidence of absence"

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]

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

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]

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.

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]

I'm not sure how to deal with the statement if it is interpreted to mean "complete absence of any kind of evidence."

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

 

[EnumaElish]

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

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]

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

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]

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.

...

So, how would one symbolize this statement?

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]

Evidence isn't really related to logic.

...

Because evidence can't prove anything, it's sort of independent of the actual truth value of a statement.

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]

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.

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]

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.

Evidence of absence counts as evidence of absence.

 

[DaveC426913]

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.

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.

 

[JeremyHorne]

Re: "Absence of evidence is not evidence of absence", I see both words being used as things and properties of individuals. "Absence" in the first part is a quality that the thing "evidence" has. In the second, "absence" is the thing that possesses the quality of being evidence. Thus, to symbolize this, you'd need something like:
e = (thing called) evidence
a = (thing called) absence
E = quality, or property of being evidence
A = quality of being absent

I'd use a conditional, ->

(A)(E)(Ae -> ~Ea) - If there is a thing called "evidence" possessing the property of being absent, the it is not the case that a thing called "absence" possesses the quality of being evidence.

This statement is only that and I do not see a real argument, unless one considers enthymemes. Then, you could do just about anything you want.

 

[SW VandeCarr]

The simplest way might be to use ordinary material implication.

Given (~P^Q) and (~P^~Q) are true then (~P^Q)V(~P^~Q) is true . That is, the absence of the antecedent (the evidence) is consistent with either the presence or the absence of the consequent.

 

[honestrosewater]

Given (~P^Q) and (~P^~Q) are true then (~P^Q)V(~P^~Q) is true . That is, the absence of the antecedent (the evidence) is consistent with either the presence or the absence of the consequent.

Given (~P^Q) and (~P^~Q) are true, everything is true. The statements are contradictory.

This immediately made me think of the fallacy of denying the antecedent, but I can't figure out a direct translation. It's obviously true on a strict interpretation (absence of evidence is not evidence by the definition of "absence"). But I think its intended interpretation is at the least misleading since the situations that it's addressing in practice concern probabilities of events, and "the absence of evidence" is actually a conditioning event. I think I have a different interpretation in mind than techmologist did, though. I take it to just be making a claim of independence, though it's certainly not clear about it.

 

[SW VandeCarr]

Given (~P^Q) and (~P^~Q) are true, everything is true. The statements are contradictory.

I take it to just be making a claim of independence, though it's certainly not clear about it.

It's (~P^Q) OR (~P^~Q). To have Q and ~Q in a conjunctive statement (AND) would be a contradiction. I believe the formula I wrote captures the essence of the idea that the absence of evidence does not necessarily imply evidence of absence. I think the insertion of "necessarily" describes the situation better than the usual form of this statement.

 

[cluebcke]

To me this problem arises only due to the brevity of the statements. What the two components amount to, if spelled out, are:

P. There is no evidence that supports the proposition
Q. There is evidence that refutes the proposition

When stated thusly, obviously Q does not follow from the truth of P.

In this particular case we're using the word "absence" in two different ways, but this isn't explicit in the terse formulation. In the case of "absence of evidence", the thing that's absent is evidence that supports some proposition.

In the following clause, "evidence of absence", the thing that's potentially absent is the subject of the proposition, not evidence in support or refutation of it.

The confusion only arises because the proposition in question, though it's never stated, is implied to be about the presence or absence of some thing in the universe.

 

[SW VandeCarr]

To me this problem arises only due to the brevity of the statements. What the two components amount to, if spelled out, are:

P. There is no evidence that supports the proposition
Q. There is evidence that refutes the proposition

I disagree with your interpretation. You say P but talk about ~P in terms of evidence. Then you mention evidence again when interpreting the consequent. Evidence is not a consequent of evidence.

What I attempted to do was to show that ~P is consistent with either ~Q or Q and such a statement is valid. It means that the absence of evidence doesn't allow one to conclude anything about the truth of the consequent.

 

[honestrosewater]

It's (~P^Q) OR (~P^~Q). To have Q and ~Q in a conjunctive statement (AND) would be a contradiction.

What is OR? The "and" that you wrote here:

Given (~P^Q) and (~P^~Q) are true then (~P^Q)V(~P^~Q) is true .

It doesn't look like OR. It looks like "and", but I am only looking at the letters. If you meant OR, then what you said is not contradictory but plainly tautological.

 

[SW VandeCarr]

What is OR? The "and" that you wrote here:
It doesn't look like OR. It looks like "and", but I am only looking at the letters. If you meant OR, then what you said is not contradictory but plainly tautological.

The two relevant premises of material implication are stated on the left (post 19). On the right I show the derived disjunctive formula based on these two individual premises. And you are correct. It is a tautology. That's the point. You can't conclude anything regarding the truth of the consequent from the absence of evidence.

Earlier in this thread there was a cookie jar example and a football field example. The idea was that an empty cookie jar is plainly decidable. But this is not the absence of evidence. The evidence is plainly there. The cookie jar is empty. In the football field example, you haven't found the cookie, but you haven't searched the entire field. Therefore cannot conclude anything about the presence or absence of the cookie.

 

[Phrak]

See the Wikipedia article "Argument from ignorance"