Not all evidence of (2) is evidence of (1). (1) and (2) are formally equivalent. 'Evidence' is a matter of fact.
A green apple is evidence in support of (2) in terms of matters of fact. It is not evidence in support of (1) in terms of matters of fact.

Not all evidence of (2) is evidence of (1). (1) and (2) are formally equivalent. 'Evidence' is a matter of fact.
A green apple is evidence in support of (2) in terms of matters of fact. It is not evidence in support of (1) in terms of matters of fact. — quine

If they're logically equivalent then how can something support (2) in terms of matters of fact (whatever that means) but not (1)?

Formally, (1) and (2) are the one same proposition. 'Evidence' is a tricky expression. Evidence is irrelevant to formal equivalence.
In natural language, a green apple is relevant to (2), but it isn't relevant to (1). Formally accurate language guarantees the equivalence of (1) and (2). Natural language merely shows that they are two different strings of symbols.

They're logically equivalent. Therefore evidence that supports the truth of one ipso facto supports the truth of the other. You can't say that you have evidence that one statement is true but not evidence that a logically equivalent statement is true.

I said that 'evidence' is a tricky expression. Evidence is not related to formalization. Equivalence is about formalization. Evidence is about matters of fact. The paradox suggested above is a mixture of two different kinds of issues. '(1) and (2) are logically equivalent' - It's about formalization. 'Evidence of (2) supports evidence of (1)' - It's about matters of fact.

I said that 'evidence' is a tricky expression. Evidence is not related to formalization. Equivalence is about formalization. Evidence is about matters of fact. The paradox suggested above is a mixture of two different kinds of issues. '(1) and (2) are logically equivalent' - It's about formalization. 'Evidence of (2) supports evidence of (1)' - It's about matters of fact. — quine

And it's still the case that (1) is true iff (2) is true. Therefore evidence that supports the truth of (2) also supports the truth of (1).

'A green apple' is not related to the equivalence of (1) and (2). The logical form of (1) can be as follows:
(1*) For every x, if x is a raven, then x is black.
Do you see that 'a green apple' is irrelevant to (1*)?
The logical form of (2) would be:
(2*) For every x, if x is not black, then x is not a raven.
Do you see that 'a green apple' is not related to (2*)?
You should be able to distinguish the talk about formalization from the talk about matters of fact.

(2*) For every x, if x is not black, then x is not a raven.
Do you see that 'a green apple' is not related to (2*)? — quine

No. Green apples are evidence that support this claim. They're not black and not ravens.

The logical form of 'there are green apples' goes as follows:
(3) There exists some x such that x is green, and x is an apple.
Do you think that (3) is relevant to (1*) and (2*) in terms of formalization?

So an albino raven is not a raven?
https://youtu.be/EfYH7a4RBAo

(2*) For every x, if x is not black, then x is not a raven.
Do you see that 'a green apple' is not related to (2*)?
— quine

No. Green apples are evidence that support this claim. They're not black and not ravens. — Michael

What's even more annoying is that intuition seems to say that a black raven is not evidence that non-black things are non ravens.

Unless someone wants to reform logic fairly drastically, the only thing to do is to agree with Hume that 'evidence' is all just habit and has no logic, and then follow Popper and @Cavacava and look to falsification.

What's even more annoying is that intuition seems to say that a black raven is not evidence that non-black things are non ravens. — unenlightened

And green apples are also evidence that all ravens are white.

'Evidence' is a tricky expression because it's ambiguous. When 'all ravens are black' is equivalent to 'everything that is not black is not a raven', it is about logical forms. 'Green apples' has nothing to do with the logical forms of 'all ravens are black' and 'everything that is not black is not a raven'. So, the paradox is a mixture of two different issues.

Back in the day, 'all' statements had no existential import, whereas 'some' statements did. Following which, one might have a bit of a get out, by denying that evidence can be for universal statements, but only against. Thus we have evidence above against all ravens being black and for 'some ravens are white'.

Thus 'all dragons breathe fire' cannot be supported by any number of fire breathing dragons, but is falsified by a single non-fire-breathing dragon. This fits with the Venn diagram approach, and also with Popper.

Back in the day, 'all' statements had no existential import, whereas 'some' statements did. Following which, one might have a bit of a get out, by denying that evidence can be for universal statements, but only against. Thus we have evidence above against all ravens being black and for 'some ravens are white'.

Thus 'all dragons breathe fire' cannot be supported by any number of fire breathing dragons, but is falsified by a single non-fire-breathing dragon. This fits with the Venn diagram approach, and also with Popper. — unenlightened

If that were the case then much of science would have to be dismissed. From a finite number of observations we infer general rules of nature that are applicable to everything of that type.

So you seem to address the problem of induction by denying induction as rational.

The reason that Popper "dissolved the problem of induction" was because he denied that theories were inferential at all, but were logical, and necessary. A theory purports a cause or reason for an event thing or outcome, so that the prediction of its prevalence, or necessity is not based upon observations of past events, but on a theory that purports to explain why things occurred as they did, and how they will occur later.

The evidence is falsification, but more than that, a theory needs to be able to make a novel falsifiable prediction. The theory needs to do more than merely ad hoc an explanation to an observation, or rely entirely on the information given in experience, but must be able to predict something is the case, or will be the case based on the implications of the theory.

If that were the case then much of science would have to be dismissed. From a finite number of observations we infer general rules of nature that are applicable to everything of that type. — Michael

I don't think so. You just have to preface your universals with 'As far as I know', or some such trope. Science is provisional, but you don't have to dismiss it. The general rule applies until you find an exception, and then progress is made. I mean nearly all ravens are black still, until someone discovers the vast antarctic flock of white ravens, or the red ravens of Mars.

I don't think so. You just have to preface your universals with 'As far as I know', or some such trope. — unenlightened

Then green apples support the claim "as far as I know, all ravens are black".

Well no. The evidence that as far as I know all ravens are black is limited to my having seen some ravens and them all being black. (That's the point of the dragons example.) It is evidence that some ravens are black, and all the one's Iv'e seen are black. At this point my having seen some non-black non-ravens falls out of contention,given that I have indeed seen some ravens, as being evidence that some non black things are non ravens, which cannot be translated in the same way,

That's not as clear as I'd like it to be. But consider, 'as far as I know all dragons breathe fire'. Given that my knowledge extends to no dragons at all, my evidence is nil, no matter how many non-fire-breathing non-dragons I have seen. 'As far as I know' needs to be a non zero distance wrt dragons or ravens, and not non-dragons and non ravens. The evidence makes the claim an existential one, or it is not evidence.

The evidence that as far as I know all ravens are black is limited to my having seen some ravens and them all being black. — unenlightened

But the point of the paradox is that this isn't the end of it. Green apples are evidence that as far as I know all things that aren't black aren't ravens. Which entails that they're evidence that as far as I know all ravens are black.

Yes, it is the end. Because scientific claims are existential claims, and in logic universal claims have no existential import. Thus the claim 'All ravens are black' can be taken as a priori. Part of the definition of raven-hood. In which case Cava's counterexample is indeed declared not to be a raven, but a short-necked swan, or some such. The scientific claim is that there are ravens, and they are all black, and being a raven is defined in some other way- shape rather than colour, say. The green apples are not evidence that there are ravens of any stripe.

Evidence can only apply to existential claims.

(1) All ravens are black. — Michael

Isn't this an inductive premise, though? In the sense that it cannot be deductively proven that all ravens are black? What if there was an albino raven, or a raven spray-painted lime green?

Since it is an inductive premise, then, we shouldn't be surprised when green apples fail to give us any substantial deductive information regarding the qualities of ravens.

The green apples are not evidence that there are ravens of any stripe. — unenlightened

I think the analysis is complete if one points out that 'there are ravens and all ravens are black' is no longer equivalent to 'There are non-black things and all non-black things are non-ravens'. And this asymmetry is the scientific escape from the logical paradox.

Because for logic, there is no problem in saying that all dragons breathe fire and no dragons breathe fire... As long as there are no dragons.. But the science of dragons is in its infancy

They're logically equivalent. Therefore evidence that supports the truth of one ipso facto supports the truth of the other. — Michael

What would the argument be for that?

What would the argument be for that? — Terrapin Station

They're logically equivalent because of the law of contraposition, and evidence for one is evidence for the other because they're logically equivalent.

I'm having trouble seeing why a green apple is evidence that
(forall x) (not black x) --> (not raven x)

Prima facie, it seems to be no evidence at all, since the claim is universal, and a single datum doesn't help us with the universal.

But I wonder if Bayes' Law can help us. Using P(A) to indicate the probability of A, and | do denote the conditional, we aim to prove that P(Black | Raven) =1, ie Probability that something is Black, given that it is a raven, is 1.
This is the same as P(~Black | ~Raven) = 1.

Now Bayes Law tells us that

P(~Black | ~Raven) = P(~Raven | ~Black) P(~Black) / P(~Raven)

Say we start with guess probabilities that there half the things in the world are ravens, and half of the things in the world are Black, and the two properties are independent (eg so that half of Ravens are Black), then observing a green apple will .................... aaargh, this seems to be leading to a dead end, and I'm late for work so I'm giving up for now.

I have a feeling that Bayes Law can somehow be used to interpret the observation of the green apple as evidence for the hypothesis, but right now I'm not getting there.

Can anyone see a way to make that argument, either using Bayes Law or something else.

If we can't make the interpretation then there's no paradox.

evidence for one is evidence for the other because they're logically equivalent. — Michael

That can't be the argument for why evidence for one is evidence for the other just in case they're logically equivalent. After all, it's just a restatement of what it's supposed to be an argument for.

Prima facie, it seems to be no evidence at all, since the claim is universal, and a single datum doesn't help us with the universal. — andrewk

The thinking is that if a black raven is supporting evidence that all ravens are black (not proof, note), then by the same token a green apple is supporting evidence that all non black things are non-ravens.

It's like love and marriage, you can't have one without the other. Unless you use unenlightened's patent interpretation of scientific generalisations.

That can't be the argument for why evidence for one is evidence for the other just in case they're logically equivalent. After all, it's just a restatement of what it's supposed to be an argument for. — Terrapin Station

I'm pretty sure it's axiomatic. If statements X and Y are logically equivalent then they have the same truth value in every model. Therefore if some A is evidence that X is true then it must also be evidence that Y is true.

Compare with "I am a bachelor" and "I am an unmarried man". Given that they're logically equivalent, evidence for one is evidence for the other.

I'm pretty sure it's axiomatic — Michael

Well that kind of sucks for a support of it.

It seems to me that there's a problem with it, and we're right back at what I've been harping on in other contexts: for one, we're ignoring the semantic content of the statements. But can we really do that when we're talking about whether evidence for one is evidence of the other?