Yup. Michael pointed this out fairly early on, IIRC.Green apples also "confirm" the universal statement "all ravens are white". — tom
You said:I covered that earlier.
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).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
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.)It's not counterintuitive it is just wrong. If green apples "confirm" "black ravens", they also confirm "white ravens".
Yes, I think we've all seen the white ravens, thanks. :DLet us not forget that the universal statement "all ravens are black" is in fact false!
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.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-white raven falsifies the hypothesis that all ravens are black. — Arkady
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).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
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.Actually, no, I think you're right. — Michael
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
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.Well, you were asking about p(HE|B), which, as I said, I had already covered. — tom
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).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
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 observation of a green apple, or a black raven - the corroborating evidence - is logically implied by H, and by K.
Yes, I take no issue with your definition.The "more formal statement" above is actually a definition of logical implication.
I don't have a problem with "all ravens are black"...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".
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