I actually contest that the indoor ornithologist is collecting information on the raven hypothesis. Not just limited information, no information, no information. See, there are two hidden statements here.

1) A raven is a type of bird.

2) There are no birds in my house.

If, indeed, we aren't certain if all ravens are birds, or if I have a parakeet, then the indoor ornithologist can contribute to the cause of science by reporting on the status of non black objects that they have lying around.

Suppose an alien offers a billion tons of gold if humanity can correctly answer the question, "are all zxcbfs black?" We ask the alien to repeat that,, because we didn't quite catch that last bit.

Then they hand every human a magic wand that detects if an object is a zxcbfs or not.

Initially, you should scan everything in your house, everything in your yard. Every result is something.

The paradox seems to flip back and forth between the two scenarios, one where we know a great deal about ravens, and one where we pretend raven is some mysterious category never before conceived by humans.

Investing pink slippers only makes sense if you think they might be a raven.

I definitely agree with you. Indoor ornithology is redundant. I presented a very simplified model where we implicitly assumed the distribution of ravens is uniform. This is of course not true (at least I hope not!). I found this paper[1] from a couple of years ago that explains it well - it redefines what constitutes as evidence with regards to random sampling and generally makes everything more rigorous.

...It seems like everyone agrees then? Where are the grounds for deep epistemological and methodological doubts? The crux of this objection is that the set of non black objects isn't well defined, and possibly infinite in size, and it is therefore impossible to study. Which seems a significant hole in the theory. The bayesian argument says that even if it were possible, the set of non black objects is very large, and the information returns become uselessly small very quickly. Looking at a single raven gives more information than all the non black household items in the world. This is another massive hole in the paradox.

Also, from a non philosophical real world perspective, real world ornithologists actually do look at non black objects to test their hypothesis. Specifically, they look at non black birds that might be ravens. More holes.

The boat leaks from all sides! Who yet sails in it? What are all those academic papers talking about, or all they all trying to poke more holes in it, like collecting hull samples from the Titanic?

You mention the king of France doubt, but Bayes only leads you astray if you set up the question incorrectly. "All kings of France are bald" indeed gets higher the more non royal French citizens you look at, which is helpful because its actually true. There are no hairy French kings.

"There is exactly one king of France and he is bald" moves in complicated ways (im assuming), but moves to very low as the last few citizens are analyzed, because we probably should have found the king already, if there was one. This matches my intuitions about probability.

I'm probably misinterpreting something. If you have a link to someone who believes that this produces serious problems for the scientific method, I would love to read them.

Concisely. An evidence e confirms a hypothesis h iff P(h/e)>P(h). Let us consider the hypothesis h=“all ravens are black” in the most current context; the value P of its probability calculated on the observation of some ravens, all black, is by far higher than the value P’ calculated on the observation of some non-black individuals, all non-ravens, though h and h’=”all non-black individuals are non-ravens” are logically equivalent. Now (Nicod’s postulate) the new evidence of a non-black non-raven individual (a red apple, say) actually confirms h’, increasing P’ up to P”; but even thus increased, P”<<P, therefore, by definition, h is not at all confirmed by the red apple, because we can already count on its by far higher value P.

Here is the winning trace for the definitive solution of Hempel’s paradox (in the full respect of classical logic and of Nicod’s postulate). I have similar traces for Goodman’s riddle, McGee’s counter-examples of Modus Ponens, Bértrand’s polyvalent geometrical probabilities, up to the general solution of logical paradoxes; in fact all these troubles arise from a wrong approach to the various problems.

Following a friendly advice, I reconsider my short paper above because, although I have no doubt that it illustrates the true solution, I have some doubt about its easy accessibility.

First of all the symbology.

U Is our universe

k is the statute, i.e. what we currrently know about U and, as such, represents the context on whose base we judge the probability of various hypotheses

h is the hypothesis (based on k) that all ravens are black

h ‘ is the hypothesis (based on k) that no non-black individual is a raven

e is a new evidence of a non-black non-raven individual (a red apple, say)

h” is the hypothesis (based on k&e) that no non-black individual is a raven

P is the probability of h based on k

P’ is the probability of h’ based on k

P” is the probability of h”

n is the statistical probability of a black plumage

m is the ratio in U between the totality of ravens and the totality of non-black individuals

Hempel’s equivalence condition states that If A confirms B, then A confirms every C logically equiivalent to B.

Nicod’s criterion states that the observation in U of an A which is B, confirms the hypothesis that in U every A is B.

Modus Tollens can be applied to whateve inductive generalization: if all As are B, all ¬Bs must be ¬A. Thus we have two distinct inductive generalizations. In the case of specific interest the first generalization looks at ravens to ascertain whether they are black or not, the second looks at non-black individuals to ascertain whether they are not or are ravens. The two generalizations are logically equivalent (if one of them is true or false, also the other one must be respectively true or false): but this does not mean at all that, given a determined context, the probability under this context of a raven beig non-black is the probability of a non-black individual being a raven.

Supposing that the probability of a black plumage is 1/n, after the observation of only ten ravens, all black, h becomes almost certainty (¬h=1/n10), while to reach more or less equal result looking at non-black individuals we must resort to an astonishing number of observations, all of non- ravens. In fact, before starting to suspect that there are not ravens among the non-black individuals one must have examined at least m individuals. But in order to reach a similar almost cerrtainty this number should be multplied by n10. And it is just this huge difference between P and P’ which makes the red apple an evidence confirming h’ (no paradox in stating that a red apple confirms, even if infnitesimally, the ypothesis according to which no non-black individual is a raven) but does not confirm h (so that an evidence e be a confirmation of a hypothesis having a given probability, the resulting probability should be greater than the given one).

Every inductive generalization is supported by a context. The current context supporting our generalization h is provided by the observation of some (prudentially; ten, but the more , the better) ravens, all black, scatterred among a huge number (prudentially; one million) of very disparate other observations.

Someone who, called to confirm empirically the hypothesis that all ravens are black, instead of displaying the black raven 1, the black raven 2, the black raven 3 and so on, displayed triumphantly a red apple, a blue tie, a white statue and so on, adducing the (anyhow right) reason the if all ravens are black, all non black individuals are non-ravens, he would reveal to be a subject to keep under psychiatrical control. Why? Because in practice the only way he choose in order to increase the probability of the inductive generalization h (whose probability is extremely higher than h’) is the displaying of black ravens.

Let me insist. Since P>>P’, even if P” is infinitesimally confirmig P’, P”<<P, therefore P” is astrally far from confirming that all ravens are black (a generalization already supported by the astrally stronger evidence grounded on the observation of only ten black ravens) .

The paradox shows that Hempel’s equivalence condition and Nicod’s criterion are incompatible. Now, since there are nearly infinite (prudentially: billions) observations which confirm P’ without absolutely confirminging P (and the observation of a red apple is one of them) the paradox is completely overcome.

I actually contest that the indoor ornithologist is collecting information on the raven hypothesis. Not just limited information, no information, no information. See, there are two hidden statements here.

1) A raven is a type of bird.

2) There are no birds in my house.

If, indeed, we aren't certain if all ravens are birds, or if I have a parakeet, then the indoor ornithologist can contribute to the cause of science by reporting on the status of non black objects that they have lying around.

Suppose an alien offers a billion tons of gold if humanity can correctly answer the question, "are all zxcbfs black?" We ask the alien to repeat that,, because we didn't quite catch that last bit.

Then they hand every human a magic wand that detects if an object is a zxcbfs or not.

Initially, you should scan everything in your house, everything in your yard. Every result is something.

The paradox seems to flip back and forth between the two scenarios, one where we know a great deal about ravens, and one where we pretend raven is some mysterious category never before conceived by humans.

Investing pink slippers only makes sense if you think they might be a raven.

I'm on a phone, so my formatting is trash, sorry.

edited Mar 1, 2022I definitely agree with you. Indoor ornithology is redundant. I presented a very simplified model where we implicitly assumed the distribution of ravens is uniform. This is of course not true (at least I hope not!). I found this paper[1] from a couple of years ago that explains it well - it redefines what constitutes as evidence with regards to random sampling and generally makes everything more rigorous.

[1] https://biblio.ugent.be/publication/8644425/file/8644426

edited Mar 1, 2022...It seems like everyone agrees then? Where are the grounds for deep epistemological and methodological doubts? The crux of this objection is that the set of non black objects isn't well defined, and possibly infinite in size, and it is therefore impossible to study. Which seems a significant hole in the theory. The bayesian argument says that even if it were possible, the set of non black objects is very large, and the information returns become uselessly small very quickly. Looking at a single raven gives more information than all the non black household items in the world. This is another massive hole in the paradox.

Also, from a non philosophical real world perspective, real world ornithologists actually do look at non black objects to test their hypothesis. Specifically, they look at non black birds that might be ravens. More holes.

The boat leaks from all sides! Who yet sails in it? What are all those academic papers talking about, or all they all trying to poke more holes in it, like collecting hull samples from the Titanic?

You mention the king of France doubt, but Bayes only leads you astray if you set up the question incorrectly. "All kings of France are bald" indeed gets higher the more non royal French citizens you look at, which is helpful because its actually true. There are no hairy French kings.

"There is exactly one king of France and he is bald" moves in complicated ways (im assuming), but moves to very low as the last few citizens are analyzed, because we probably should have found the king already, if there was one. This matches my intuitions about probability.

I'm probably misinterpreting something. If you have a link to someone who believes that this produces serious problems for the scientific method, I would love to read them.

Soluttion of Hempel’s paradox.

Concisely. An evidence e confirms a hypothesis h iff P(h/e)>P(h). Let us consider the hypothesis h=“all ravens are black” in the most current context; the value P of its probability calculated on the observation of some ravens, all black, is by far higher than the value P’ calculated on the observation of some non-black individuals, all non-ravens, though h and h’=”all non-black individuals are non-ravens” are logically equivalent. Now (Nicod’s postulate) the new evidence of a non-black non-raven individual (a red apple, say) actually confirms h’, increasing P’ up to P”; but even thus increased, P”<<P, therefore, by definition, h is not at all confirmed by the red apple, because we can already count on its by far higher value P.

Here is the winning trace for the definitive solution of Hempel’s paradox (in the full respect of classical logic and of Nicod’s postulate). I have similar traces for Goodman’s riddle, McGee’s counter-examples of Modus Ponens, Bértrand’s polyvalent geometrical probabilities, up to the general solution of logical paradoxes; in fact all these troubles arise from a wrong approach to the various problems.

If some reader is interested in the matter, write to me at italo@italogandolfi.com.

Following a friendly advice, I reconsider my short paper above because, although I have no doubt that it illustrates the true solution, I have some doubt about its easy accessibility.

First of all the symbology.

U Is our universe

k is the statute, i.e. what we currrently know about U and, as such, represents the context on whose base we judge the probability of various hypotheses

h is the hypothesis (based on k) that all ravens are black

h ‘ is the hypothesis (based on k) that no non-black individual is a raven

e is a new evidence of a non-black non-raven individual (a red apple, say)

h” is the hypothesis (based on k&e) that no non-black individual is a raven

P is the probability of h based on k

P’ is the probability of h’ based on k

P” is the probability of h”

n is the statistical probability of a black plumage

m is the ratio in U between the totality of ravens and the totality of non-black individuals

Hempel’s equivalence condition states that If A confirms B, then A confirms every C logically equiivalent to B.

Nicod’s criterion states that the observation in U of an A which is B, confirms the hypothesis that in U every A is B.

Modus Tollens can be applied to whateve inductive generalization: if all As are B, all ¬Bs must be ¬A. Thus we have two distinct inductive generalizations. In the case of specific interest the first generalization looks at ravens to ascertain whether they are black or not, the second looks at non-black individuals to ascertain whether they are not or are ravens. The two generalizations are logically equivalent (if one of them is true or false, also the other one must be respectively true or false): but this does not mean at all that, given a determined context, the probability under this context of a raven beig non-black is the probability of a non-black individual being a raven.

Supposing that the probability of a black plumage is 1/n, after the observation of only ten ravens, all black, h becomes almost certainty (¬h=1/n10), while to reach more or less equal result looking at non-black individuals we must resort to an astonishing number of observations, all of non- ravens. In fact, before starting to suspect that there are not ravens among the non-black individuals one must have examined at least m individuals. But in order to reach a similar almost cerrtainty this number should be multplied by n10. And it is just this huge difference between P and P’ which makes the red apple an evidence confirming h’ (no paradox in stating that a red apple confirms, even if infnitesimally, the ypothesis according to which no non-black individual is a raven) but does not confirm h (so that an evidence e be a confirmation of a hypothesis having a given probability, the resulting probability should be greater than the given one).

Every inductive generalization is supported by a context. The current context supporting our generalization h is provided by the observation of some (prudentially; ten, but the more , the better) ravens, all black, scatterred among a huge number (prudentially; one million) of very disparate other observations.

Someone who, called to confirm empirically the hypothesis that all ravens are black, instead of displaying the black raven 1, the black raven 2, the black raven 3 and so on, displayed triumphantly a red apple, a blue tie, a white statue and so on, adducing the (anyhow right) reason the if all ravens are black, all non black individuals are non-ravens, he would reveal to be a subject to keep under psychiatrical control. Why? Because in practice the only way he choose in order to increase the probability of the inductive generalization h (whose probability is extremely higher than h’) is the displaying of black ravens.

Let me insist. Since P>>P’, even if P” is infinitesimally confirmig P’, P”<<P, therefore P” is astrally far from confirming that all ravens are black (a generalization already supported by the astrally stronger evidence grounded on the observation of only ten black ravens) .

The paradox shows that Hempel’s equivalence condition and Nicod’s criterion are incompatible. Now, since there are nearly infinite (prudentially: billions) observations which confirm P’ without absolutely confirminging P (and the observation of a red apple is one of them) the paradox is completely overcome.