Just curious. What happens to the millions of "confirmations" that "all ravens are black", by observing not only black ravens but literally every non-black non-raven thing you have ever seen, when you encounter these?

Green apples also "confirm" the universal statement "all ravens are white". — tom

Yup. Michael pointed this out fairly early on, IIRC.

I covered that earlier.

You said:

K is compatible with any evidence. p(E|K) is still 1, and (KE) and (K) have the same truth value. I certainly seems weaker, but I can see no reason that K does not logically imply E, just as it implies not(E).

You might regard this as a more formal statement:

K => E iff p(E|KB) = 1 for every B

My original concerns stand. K is a hypothesis which is supposedly compatible with "any evidence," which is completely at odds with its being falsifiable (indeed, this seems to smuggle your conclusion into the proof itself, thereby begging the question).

Also, I still suspect you're making an illicit move in proposing that if P(E|K) = 1, that P(K&E) = P(K). Unless K & E are both necessary truths, then the probability of their conjunction must be less than either conjunct alone (though, they probably should not be considered as statistically independent, so this point is debatable. Either way, I'm skeptical that E drops out so smoothly from the equations).

It's not counterintuitive it is just wrong. If green apples "confirm" "black ravens", they also confirm "white ravens".

Yes, but evidence can be consistent with multiple hypotheses, which is called underdetermination, and is well-known in the philosophy of science. (I think that most treatments of this problem, even when they allow that non-black non-ravens confirm the hypothesis, treat such observations as very weak evidence.)

My problem was in saying that a given hypothesis and its negation should entail the same evidence. If H is "all ravens are black," and K is "not all ravens are black," K is not only amenable to confirmation, it is verified by a single instance of a non-black raven. Clearly, whatever we may think about the possibility of universally-quantified hypotheses being confirmed, an observation of a non-black raven decidedly does not confirm "all ravens are black."

Let us not forget that the universal statement "all ravens are black" is in fact false!

Yes, I think we've all seen the white ravens, thanks. :D

Just curious. What happens to the millions of "confirmations" that "all ravens are black", by observing not only black ravens but literally every non-black non-raven thing you have ever seen, when you encounter these? — tom

A hypothesis can be confirmed by evidence but still turn out to be false. "Confirmation" is not equivalent to "verification." The observation of a single non-black raven falsifies the hypothesis that all ravens are black.

A hypothesis can be confirmed by evidence but still turn out to be false. "Confirmation" is not equivalent to "verification." The observation of a single non-white raven falsifies the hypothesis that all ravens are black. — Arkady

So, what happens to the millions of "confirmations"?

And, why am I still seeing them? I can literally look at green apples whenever I want.

So, what happens to the millions of "confirmations"? And, why am I still seeing them? I can literally look at green apples whenever I want. — tom

Not sure what you're asking here. The apples are no less green because there are white ravens. We can accumulate evidence for a hypothesis which later turns out to be false (were this not the case, there would in fact be no difference between "justification" and "truth," at least with regards to empirical hypotheses).

Actually, no, I think you're right. — Michael

It gets worse: every egg-sized block of empty space we observe also confirms the hypothesis that "all eggs are white" (again, assuming that we've already observed at least one white egg), because that is one less egg-sized block of space which could potentially contain a non-white egg.

You said:

K is compatible with any evidence. p(E|K) is still 1, and (KE) and (K) have the same truth value. I certainly seems weaker, but I can see no reason that K does not logically imply E, just as it implies not(E).

You might regard this as a more formal statement:

K => E iff p(E|KB) = 1 for every B

My original concerns stand. K is a hypothesis which is supposedly compatible with "any evidence," which is completely at odds with its being falsifiable (indeed, this seems to smuggle your conclusion into the proof itself, thereby begging the question). — Arkady

Well, you were asking about p(HE|B), which, as I said, I had already covered.

The observation of a green apple, or a black raven - the corroborating evidence - is logically implied by H, and by K.

The "more formal statement" above is actually a definition of logical implication.

If you don't like the "not all ravens are black" fro some reason, then change it. There are several others you could chose, "all ravens are black except the white ones", "all ravens are black or white".

Well, you were asking about p(HE|B), which, as I said, I had already covered. — tom

I know, but I had issues with your treatment of both H and K, which i will discuss in more detail below. In the meantime, I offer this correction to one of my points.

Also, I still suspect you're making an illicit move in proposing that if P(E|K) = 1, that P(K&E) = P(K). Unless K & E are both necessary truths, then the probability of their conjunction must be less than either conjunct alone (though, they probably should not be considered as statistically independent, so this point is debatable. Either way, I'm skeptical that E drops out so smoothly from the equations). — Arkady

I added bolding to my above quote, because the non-statistically independent nature of K and E (assuming that K entails E) is in fact the key here, at least according to some quick and dirty refresher research I did. I can flesh out my point, if need be, but suffice to say, I now agree that, if K entails E, then P(K&E) = P(K).

The observation of a green apple, or a black raven - the corroborating evidence - is logically implied by H, and by K.

This is part of the sticking point. I don't see how K (i.e. "not all ravens are black") implies the observation of a black raven. It is at most logically consistent with this observation. More generally, I don't see how a statement and its negation both imply the same thing (at least with regards to empirical hypotheses).

The "more formal statement" above is actually a definition of logical implication.

Yes, I take no issue with your definition.

If you don't like the "not all ravens are black" fro some reason, then change it. There are several others you could chose, "all ravens are black except the white ones", "all ravens are black or white".

I don't have a problem with "all ravens are black"...

Has anyone here read any existing literature on the subject? There's lots!

First it should be noted that nothing about evidence and confirmation is necessitated by classical logic, simply because these concepts do not belong in classical logic. That's not to say that a theory cannot be built on deductive foundations (that's what Hempel, who came up with the paradox, as well as a number of others, attempted to do). However, even with classical logic as a background there are various ways of going about it, and different models and starting assumptions will yield different results. And then there are various non-deductive theories of confirmation: Bayesian and even more exotic theories, such as two-parameter models. (A Popperian will just dismiss the challenge, since according to her there is no such thing as confirmation. And that's why few pay attention to Popperians :P)

If the claim is that "all ravens are black", then it's true to say that every new raven I see that's black supports the claim. If that's true then yes, by equivalence it's also true to say that if I see a non-raven that's not black this also supports the claim.

However the fact that they both support the claim, does not mean that they provide equal support.

Consider what it would take to be certain that all ravens are black, one of the following needs to occur.
i) ravens need to be defined as being black
ii) we need to see all the ravens and all of them need to be black
iii) we need to see all the non-black objects, none of them can be ravens and I need to know ravens exist (I need to state that ravens exist here otherwise you could prove that all unicorns are pink by observing that all non-pink objects are not unicorns).

Case i is trivial. Case ii requires the observation of far fewer objects than case iii. Therefore one observation of a black raven supports the claim more than the observation of a non-black non-raven. The degree that an observation of a non-black non-raven supports the claim depends on how many non-black objects there are. If there are a finite number of non-black objects, then an observation of a non-black non-raven would support the claim some non zero amount. If you claim there are an infinite amount of non-black objects then the support for the claim from observing a non-black non-raven would be infinitely small. In either case, it would be hard to imagine any number of observations of non-black non-ravens changing your stance on whether all ravens are black.

We ought to be suspicious of the equivalence of (1) and (2), since we can find cases where truth value judgments about them don't coincide.

For example, suppose there are no ravens. Then (2) is true, but the status of (1) is uncertain.

I have savored many paradoxes, and I have passioned an aeipathy for ravens.

If an item or idea shares an absolute trait with its fellows, and without exception, then we can reasonably assume that everything which is lacking in that trait is not of that designation. We may do this by the process of elimination, categorization, and exclusion.

The Raven Paradox is highly conditional, as thre are few objects which share such an absolute similarity. When items are compared, such as Johnny Truants and black ravens, it is natural to seek evidence for the conclusion. When items other than ravens are mentioned, they give evidence in the context of the statement. I am not certain that this oddity should be considered a veritable paradox. In considering the problem, we should remember that abstractions between divers objects are a common construct of our existence. Exclusions are a crucial portion of evaluation. While the problem is counterintuitive, it may seem an oddment only according to our wonted schematics of awareness.