Philosophy: How can we be sure that all crows are black?

Does the discovery of a white shoe in the yard confirm the hypothesis that all crows are black? This question is of great importance in the philosophy of science, and the answer to it is not so simple 

Marius Cohen, "Galileo" magazine
It cannot be doubted that modern science has brought about an enormous growth in human knowledge in the last few centuries. At the same time, the principles on which science rests in its attempt to decipher the secrets of nature have often been called into question. The paradox discussed below constitutes a critical examination of some of these principles.
Our ability to recognize regularities in nature is very important: we learn in which seasons of the year we can grow different agricultural crops, which materials are most suitable for making tools, clothes or houses, which animals are dangerous to us, which foods satisfy us, which of our acquaintances are good for us and which are bad, etc. in these
Our ability to recognize regularities is based on a relatively simple cognitive mechanism: if we notice a recurring phenomenon (for example, we discover in our early childhood that unsupported objects fall to the ground), we tend to assume that this is a general phenomenon (that is, everything that is not supported falls to the ground). Every repetition of the phenomenon strengthens our confidence in the hypothesis, while every deviation from it (for example, a balloon filled with helium or the holiday plane in the sky) leads us to understand that it is wrong, and then we can try and come up with a new hypothesis, suitable for the observations.

The paradox
Similarly, science also makes hypotheses based on repeated phenomena, where any observation that matches the hypothesis is seen as confirming it, while an observation that does not match it is seen as disproving it. For example, if zoologists and amateur birders have only observed black crows to date, it would be reasonable to assume that all crows are black.
Every other black crow we notice will confirm this hypothesis, but if one day a crow of a different color is discovered, our hypothesis will be disproved.
The principle that particular instances of a general proposition confirm it is called Nicod's principle, named after the French philosopher and logician Jean-Georges Pierre Nicod, who proposed this principle at the beginning of the 20th century. On the face of it, this principle seems plausible: if we hypothesize (after noticing this several times), that after every lightning there is a sound of thunder, then every such case, of lightning accompanied by thunder, will confirm our hypothesis and increase our confidence in it.
A second principle that we tend to accept is based on the concept of logical equivalence, and is therefore called the principle of equivalence. Two propositions are called logically equivalent if they both necessarily have the same truth value, that is, when one of them is true so is the other, and when one of them is false so does its equivalent. For example, the two claims "if it is 12 noon, there is light outside" and "if there is no light outside, it is not 12 noon" are valid claims, because if one of them is true, so is the other.
It is easy to see that any claim of the type "if A then B" is equivalent to the claim "if not B then not A". Similarly, the two claims "all A is B" and "all that is not B is not A" are equivalent. For example: the claim "all ravens are black" is equivalent to the claim "anything that is not black is not a crow", since the truth of one of the claims necessarily entails the truth of the other claim. According to the principle of equivalence, any case that confirms a certain claim necessarily also confirms any claim that is equivalent to it. After all, if our confidence in the hypothesis that all crows are black increases, our confidence in the hypothesis that everything that is not black is not a crow necessarily increases, and vice versa.
However, the combination of Nico's principle with the equivalence principle brings us to a surprising conclusion: according to Nico's principle, the cases that are supposed to confirm the hypothesis "everything that is not black is not a crow" are things that are not black, for example: a white shoe, a red box, a yellow pencil, etc. However, according to the principle of equivalence, all of these should also confirm the equivalent claim, which is: "All crows are black". It follows from this that if we find a white shoe in the yard of our house, it will be a confirmation of the hypothesis that all crows are black!
points for thinking
This paradox stems from the fact that reasonable assumptions (Nico's principle and equivalence principle) lead in a reasonable logical course to an unreasonable conclusion (that a white shoe confirms the hypothesis that all crows are black). If so, in order to get out of the tangle we have fallen into, we must give up at least one of the assumptions, or doubt the legitimacy of the logical process, or accept the conclusion, no matter how improbable it may be. Which solution direction would you choose in this case?
If you saw a white bird in the distance, and when you got closer you discovered that it was not a crow but a dove, would you be willing to see this as confirmation of the hypothesis that all crows are black (perhaps in contrast to the white shoe you found in the yard)?
Does the fact that the number of non-black things in the world is significantly greater than the number of crows seem relevant to our case?

Solution directions for the paradox
Hempel himself was ready to accept the conclusion, despite its improbability. According to him, the degree of confirmation of a black crow for the hypothesis that all crows are black (and the claim equivalent to it) is significantly greater than the degree of confirmation of a white shoe for the claim that everything that is not black is not a crow (and the claim equivalent to it), and this is for the reason that the number of crows in the world is significantly smaller than the number of things that are not black. If the number of non-black things in the world (or for that matter - in a large hall) were the same as the number of black crows in it, then everything that is not black and is not a crow would confirm both equal claims to the same extent that a black crow would.
Not everyone accepted Hampel's position, and there were those who chose to attack at least one of the principles underlying the paradox. The Niko principle, for example, is considered particularly problematic. Does every case that matches the hypothesis actually confirm it? For example, if we hypothesize that there are no mice in our house (or in the more cumbersome formulation: all the mice are outside our house), would a mouse we discovered outside the house confirm the hypothesis? After all, we can find any number of mice in the street or in a field, and we still won't know if there is a mouse in the house or not. Therefore, there are those who are willing to give up the principle of cleanliness, and claim that finding a white shoe does not confirm the hypothesis that everything that is not black is not a crow.
There are also those who criticize the principle of equivalence, and claim that the scientific process of confirming hypotheses must be separated from logical inferences. Therefore, say the adherents of this position, a white shoe does confirm the hypothesis that everything that is not black is not a crow, but not the hypothesis that all crows are black. This is even though the claims are logically sound and the truth of one of them entails the truth of the other.
This last position, although some advocate it, is not particularly popular, and this is because in scientific investigation, logical inferences are always involved, and it seems that by removing them from the rules of the game, we will lose more than we will gain. The waiver of the Niko principle is more acceptable as a solution to the paradox of the crows, but it seems to be "throwing the baby out with the bathwater" here. There is no doubt that the Nico principle requires corrections, but it would be a shame to give it up completely, since it is an important tool both in everyday thinking and in scientific work.

How can we, then, improve it?
It would be reasonable to demand that only a case capable of refuting the hypothesis could also confirm it. To disprove the hypothesis that there are no mice in the house, one must look for them in the house and not outside it. If a mouse is indeed found in the house, this will disprove the hypothesis, so as we continue to search the house without finding a mouse, we will confirm it. On the other hand, looking for mice outside the house cannot lead to refuting the hypothesis, and therefore cannot lead to confirmation either.
Under this new constraint that we added to Nico's principle, we will prove that something that is not black may confirm the hypothesis that everything that is not black is not a crow only under certain conditions: if we identify a shoe in the yard, then this fact does not disprove the hypothesis that everything that is not black is not a crow (because only a crow A person who is not black can disprove it, and since we know that it is a shoe, this condition is not met.) On the other hand, if we see an unidentifiable white bird from a distance, then for us this bird may also be a crow, thus disproving the hypothesis.
When the bird approaches us, and it seems that it is nothing but a pigeon, it will be for us a confirmation of the hypothesis that everything that is not black is not a crow. That is, in any situation where we notice something that is not black, and according to all the information we have, it may be a crow, then its final identification as something that is not a crow will confirm the hypothesis that everything that is not black is not a crow, and the same goes for the logically equivalent hypothesis, which is that all crows They are black (one can imagine a scenario where even a white shoe might provide confirmation of the hypothesis).
If so, we returned to Hempel's original proposal, but we added to the Nicho principle the constraint that only a case that has the potential to disprove the hypothesis can also confirm it. Also, in this solution we adopt Hempel's position that due to the small amount of black crows in relation to things that are not black, a black crow confirms the hypothesis to a greater extent than a thing that is not black (subject, of course, to the condition we added). This does not harm the principle of equivalence, because a black crow equally confirms the two equal claims, and something that is neither black nor a crow confirms both equally (although significantly less than a black crow).

Dessert
The hypothesis that there is no person whose height reaches 3 meters is a hypothesis that is confirmed every day in our encounters with people. The hypothesis equivalent to this hypothesis is the hypothesis: everything that is 3 meters tall or higher is not a person. Is the hypothesis that there is no person whose height reaches 3 meters confirmed every day also by the tall buildings that come our way? Can you think of a scenario where a tall building would actually confirm the hypothesis?

From the magazine "Galileo"