A black raven, a white shoe and a cat.

The raven paradox, created by Carl Gustav Hempel, is a problem for the scientific community in order to establish knowledge. It starts with the next two hypotheses:
All ravens are black.
If something is not black, then is it not a raven.

These sentences are logically equivalent to each other, meaning that where one is true, the other is true and where one is false, the other is false.

In the first part of this blog I will describe and explain the raven paradox, in the second part I will propose a solution.

Importance of the paradox

The above mentioned sentences portray a big problem in the field of knowledge. The first hypothesis can be investigated to be true and for sure, it can be falsified. If one raven is found anywhere in the world with another colour than black, then that hypothesis is falsified. To achieve more knowledge in the science, one has to be able to falsify or corroborate hypotheses.
But now, let's take a look at the second hypothesis. How can it help us to get knowledge about black ravens? Well, if one sees a white shoe, then is this hypothesis corroborated, as a white shoe is not black and not a raven. But, when this hypothesis is corroborated and it is logically equivalent to the first hypothesis 'all ravens are black', then the first hypothesis is corroborated as well. Hence, one can corroborate the hypothesis 'all ravens are black' by introducing a white shoe.
This is not a very convincing method to create knowledge. This is intuitively highly unsatisfactory, yet logically impeccable. How does one solve this problem?

Gathering extra data

First I created some more hypotheses in order to avoid looking only at one example and get a grasp on what is really going on. It is so highly intuitive that I got the feeling that it is almost certain that something must have been overlooked over the years. So I added the next hypotheses to the original one:
all swans are black
if it is not a black thing, then it is not a swan

all cats are mammals
if it is not a mammal, then it is not a cat

all butterflies were once caterpillars
if it is not a catterpillar, then it will not become a butterfly

all unicorns are black
all non black things are not unicorns.

all unicorns are mammals
if it is not a mammal, then is it not a unicorn.

All 10 statements share the same venn diagram as depicted below.

venn diagram Raven paradox

The little circle is the first object, like a raven, cat or unicorn. The big circle represents the other objects or features like black, caterpillars and mammals. The rest of the space is everything that is not belonging to the second type of object. For instance everything which is not black or which is not a mammal.
This shows in another way that they are logically equivalent and that from a sole logical perspective there can't be any distinction made between those five hypotheses. One of the requirements of Hempel was therefore that one should not make use of contextual information. That is strictly logical and understandable.

The first added hypotheses are known to be false, the second and third one true, the last two added are unknown to be true or false. As long as it is unclear if unicorns do exist, is it impossible to know anything about them with certainty.
However, in all cases can be said from a strict logical point of view that a white shoe corroborates any of those statements, implying that every first will be corroborated as well. A white shoe is therefore capable, given the proper formulation, of corroborating any statement whether if it is true, false or undeterminable. Given the proper formulation it does not have the capability to falsify anything either. Based on the observation of a white shoe it is impossible to know if there are only black swans or that there are as well swans with another colour. It just corroborates.
But what does it corroborate? The original hypothesis in an indirect manner? The observation that there is more in the world than the objects which are the subjects of the original hypothesis?

My solution

Several solutions have been proposed over the years. All solutions so far are located within the realm of logic. I will not present them here, because I believe that my solution relocates the problem outside of that realm.

The puzzling thing is that the original statement and its counterpositive share their truth values completely. A statement and its contrapositive are logically equivalent, which implies that they have the same set of truth and false values. This is old knowledge in logic and beyond dispute.
However, the situation the raven paradox describes entails more than a purely logical situation. It is not about the validness of the logical formulas involved, it is about corroboration and the growth of knowledge. Although the latter might be restricted by proper use of logic, the growth of knowledge is not restricted to the realm of logic itself. It requires a context in which it takes place.
This is the reason that I prefer not to show the reasoning of the logicians. They restricted themselves to reasoning within logic, where, in my opinion, the core of the paradox, and therefore also the solution, lies out of the realm of logic. It seems to deal with a statement and its counterpositive, but it is about the value of corroboration and the possibility of growth of knowledge.

Reading their solutions I understood that logic could not solve the problem. Therefore I first gathered some more information, although I wanted to keep in mind that it all should stay equal to the original statement of the raven paradox. It would not be the first time if reformulating a problem creates another problem. Solving that problem instead of the original one would be an unfortunate straw man fallacy. Creating/collecting other statements however, helped me realizing what was the core of the problem. The hypothesis of the cat and the mammal is an example of a statement that has not the white shoe issue. Although one could say that a white shoe could be used as a corroboration, the relationship between being a cat and a mammal is a different type of relationship then between being a raven and black. It is this other type of relationship which makes the corroboration done by a white shoe futile. After some more thinking I realized there was even a third type of relationship. All three types of relationships between the two objects are shown below:

  1. has a
  2. is a
  3. implicates

ad 1. A raven is black. Being black is not an unconditional charateristic of a raven – as there can be albino ravens. You even might say that the characteristic of the colour black comes partially from the limits of our eyes to perceive colours. If we could discriminate more colours in the dark spectrum, we might have called ravens blue instead – as I perceive them as blue black and not for instance red black or green black. Green black is what I think to see on a magpie.
ad 2. A cat is a mammal. It is impossible for a cat not to be a mammal. First it is a mammal and then a cat. There are some characteristics that define a mammal. A cat has them all. Not all cats obviously as there are a lot of male cats, but the point is that there can be a hierarchy in types of objects. An 'is a'-relationship is about defining characteristics and settling a hierarchy in types of objects. For all statements in which the more specific object is said to be also of the more general type of object, this is a good description of the relationship between those objects.
ad 3. A caterpillar becomes a butterfly. There will be always more caterpillars then butterflies as all the caterpillars in the world precede the existence of any butterfly (and some caterpillars die before they could evolve into a butterfly). Likewise, every building that has four floors has a first floor, not the other way around.

The first type of relationship describes an accidental relationship between two objects. It is like individual characteristics of an object or any implementation of a class like that ravens tend to be black or each side of a square happens to be 2.
The second type of relationship is like the description of a class, that which every object, every implementation of a class, has in common, or what is defining a certain type of object.
The third type of relationship is when one object precedes, implicates or otherwise causes the existence of the other object. Like a caterpillar a butterfly and yesterday today.

Only the first type of relationship, the one with the non defining relationship creates the problem of the raven paradox, because the second type of object is not an essential part of/for the first type of object. You can't generalize from an accidental relationship, where you can from a defining feature or a causal relationship.
Yes, a white shoe is not a mammal, but because we know that all cats are supposed to be mammals, is it highly irrelevant trying to corroborate the statement 'if it is not a mammal, then is it not a cat' using a white shoe. The only type of objects that are interesting to see given this statement are mammals and cats. It doesn't give any more information if a thing outside the definition of a mammal is represented as evidence. Unless it is a cat that is not a mammal. Therefore are only mammals and cats relevant and actually only cats. The only object that could falsify this hypothesis is a cat that is not a mammal. Exactly the situation one wanted to have for the growth of knowledge.

A similar line of argumentation can be provided for the third type of relationship. Whatever is out of scope of the causal relationship is irrelevant. Although sceptics might dispute this, but it is logically valid to make generalizations using implications.
In both relationships 'is a' and 'implicates' is the direction of the argument important. Not all mammals are cats. One can not generalize in that direction. Nor can it be done from the effect to the cause. 'All caterpillars will be butterflies' is not a true statement.

Conclusions

When growth of knowledge is the problem, then the sole use of logic is not the answer as growth of knowledge is more then only applying logic. Without logic however, this problem would never get addressed properly, because with the use of logic one can make these strange observations of the unintuitive corroborations and it opens the door to further examining which type of relationships between objects can be generalized meaningfully and which not.
Generalizations are allowed when based on hierarchal information like definitions of mammals and cats and can only be done in the direction to the more general type of object. Generalizations are also allowed when based on implications/causal relationships. They go from cause to effect, not the other way around. Finally, generalizations are not allowed when based on non defining characteristics. This gives rise to the next observations and questions:

  1. one can not make valid generalizations based on non defining characteristics like for instance skin color or date of birth.
  2. can one make generalizations based on self declared identities like belonging to a religion? Can an accidental characteristic get the necessity of a defining characteristic when a person declares it as such?
  3. is it possible to make valid generalizations between non defining characteristics within an implementation? Or is any white shoe then again corroborating the relationship?
  4. is it possible that information that was supposed to be accidental/unstructured turns out to be structured data after all? Like the date of birth of a person in case astrology turns out to be true?


Used literature:
https://en.wikipedia.org/wiki/Raven_paradox
https://plato.stanford.edu/entries/confirmation/
https://explorable.com/raven-paradox
https://philosophynow.org/issues/19/Resolving_Hempels_Raven_Paradox