will the sun rise tomorrow The induction problem

In inductive inference the conclusion may be false even if the premises are true. Can we justify our belief in the conclusion that the sun will rise tomorrow just based on the fact that it has risen every morning so far? Marius Cohen is looking for a solution to the problem.

Marius Cohen, "Galileo"

A distinction must be made between two main types of inference: deductive inference and inductive inference. A deductive inference is an inference that the truth of its premises requires the truth of its conclusion. Let's look, for example, at this argument:

Assumption 1: There is no water on the planet Hema.
Assumption 2: without water there is no life.
Conclusion: There is no life on the planet Mercury.

It is of course possible that not all the assumptions of the argument are true: perhaps in the future we will discover water reservoirs under the surface of the planet, and perhaps in the more distant future we will discover planets where life exists on a different basis than water (ammonia, for example). However, the validity of this argument does not depend on the truth of its premises: if the premises are true, then the conclusion must also be true. Or in other words: it is not possible for the premises of the argument to be true and its conclusion to be false.

An inductive inference, on the other hand, is an inference whose premises lead to its conclusion with high probability, but not absolute certainty, and is characterized by generalization based on known private cases. For example, in the argument:

Assumption: all the swans observed to date have been white.
Conclusion: The next swan we see will be white.

The truth of the premise does not require the truth of the conclusion (indeed, black swans have been discovered in Australia), or in other words: it is possible that the premises of the argument are true and its conclusion false. Nevertheless, we make a lot of use of inductive inference: based on our past experience, we expect the sun to rise tomorrow morning as well; Since we got wet in the past when it rained, we will take an umbrella with us on a rainy day; When lightning flashes we expect thunder to follow; We fear another war because wars have always been bloody; and so on and so on.
It seems, therefore, that even in the inductive inference the conclusion is not certain, it is reasonable enough so that we can claim knowledge and not just a true (but unjustified) belief: we know that the sun will rise tomorrow too, we know that without an umbrella we will get wet in the rain, we know that after the lightning will come the thunder and we We know, unfortunately, that the next war will also be bloody.

The induction problem
It is appropriate to ask what justifies our (true) belief that the sun will rise tomorrow morning, if this fact does not follow deductively (and therefore surely) from the fact that the sun has done so countless times in the past? After all, it is possible that an event on a cosmological scale will occur during the night, and that the sun will not rise again. In light of this fact it might be more correct to say that we believe the sun will rise tomorrow, not that we know it;

Perhaps we should say that we believe that if we go out without an umbrella we will get wet in the rain, and not that we know it. After all, even a chicken may conclude based on past experience that the rooster came to feed it, even though this time he came to take it to slaughter (example of the 20th century English philosopher Bertrand Russell); Even if in ten consecutive throws of a normal game die the number "1" came out (the chance of this is very low, but it is not impossible), there is no justification to conclude that the next throw will also come out "1" (the chance of this is 1:6).

But our intuitions and the way we use language teach us that we attribute to many inductive inferences a high enough logical strength to justify our belief in their conclusions. This is why we think it is more correct, for example, to say: "I know that if I don't take an umbrella I will get wet in the rain" than "I believe that if I don't take an umbrella I will get wet in the rain" (which implies that the speaker is not sure that this will happen).

The assumption that these inferences have high logical strength, meaning that the conclusion follows from the assumptions with high probability, is seen as adequate justification for our belief in the conclusion, and therefore we consider this belief to be knowledge (provided it is true, of course). It is therefore possible to add to an inductive inference another assumption that expresses our belief that the inductive inference has logical strength, so that every such inference will have this structure (the assumption we added is assumption 2):
Assumption 1: All previous instances of A were of B character.
Assumption 2: The next case of A will be similar in nature to the previous cases of A.
Conclusion: the next case of A will also be of B nature.

for example:

Assumption 1: Until today, every lightning was accompanied by thunder.
Assumption 2: The next lightning will be similar in nature to the previous lightnings.
Conclusion: The next lightning will also be accompanied by thunder.

The addition of premise 2 to any inductive argument does justify accepting its conclusion. But what justifies our belief that the next case will indeed be similar to the previous cases? It is not a logical necessity (such as that in deductive inference) that requires the next case to be of the same nature as the previous case (and see the example of the chicken and the example of the cube above).

A reasonable possibility is to argue that, although there is no logical necessity here, experience shows us that this is often the case. But this is a circular argument, since we justify the problematic assumption on the basis of this assumption itself! To show this we will present the structure of the argument:

Assumption: experience teaches us that the following case is similar in nature to previous cases of the same type.
Conclusion: The next type of events that we will examine is also endowed with this feature.

When "this feature" refers to the feature formulated by assumption 2 above: the next case of this type of event will be similar in nature to its previous cases. But this is an inductive argument, and in order for its conclusion to indeed follow from its premise, we must add to it premise 2 itself (in a wording adapted to the concepts that the argument makes use of):

Assumption 1: experience teaches us that the next case is similar in nature to previous cases of the same type.
Assumption 2: The next type of events we will examine will be similar in nature to the previous types.
Conclusion: The next type of events that we will examine is also endowed with this feature.

And assumption 2 is nothing but the conclusion itself! Although a circular argument is a valid argument (it is impossible for its premises to be true and its conclusion false), it is not a good argument (and is called a failure in professional parlance): we cannot assume what we want to prove! If so, it is impossible to justify our arts with the conclusions of inductive inferences based on past experience, and the question arises whether we have another way to justify them.

The importance of the problem
The problem of induction is not a marginal philosophical problem, if only due to the fact that modern science is based on inductive inferences. Scientists do not derive the laws of nature in a logical deductive manner from any initial principles, but follow natural phenomena and conduct experiments, in an attempt to generalize the results of their observations and experiments to laws and principles that apply to the entire world, and not only to the objects of the observations and experiments.

Thus, for example, Johann Kepler, a German mathematician and astronomer of the 16th and 17th centuries, based himself on his own observations and that of Tycho Brahe, the Danish astronomer of the 16th century, to formulate three laws that characterize the movement of the stars - You went around the sun (known today as Kepler's laws). These laws did agree with Braha's observations, but what guarantees that they apply to all the celestial bodies orbiting another celestial body (for example, in another solar system)?
Isaac Newton, the famous English mathematician and physicist, showed, about eighty years later, that it is possible to derive Kepler's laws from a number of fundamental laws of mechanics (three laws of motion and the law of universal gravitation). Such a derivation is of course a deductive process, and therefore the truth of Newton's laws guarantees the truth of Kepler's laws, but Newton's laws themselves are nothing more than a generalization of the results of many experiments (that is, they were obtained through an inductive process), so what guarantees their universal validity?

At the beginning of the 20th century, Einstein showed that Newtonian mechanics is nothing but a good approximation to the laws of nature, but the problem of induction is also relevant to the theory of relativity, which replaced Newtonian mechanics, as well as to any other scientific theory. Repeated observations and experiments confirm such theories, but they cannot prove them, and there is always the possibility that the range of applicability of a theory is limited, and the next experiment will disprove it as a general theory.

A possible direction for a solution
There is no general agreement on a solution to the epistemological problem, that is, to the question of the justification of our arts in the conclusions of inductive inferences (Day himself claimed that such arts are not rational, but the result of habit). Despite this, there is one direction that deserves to be examined as a possible solution.

Let's take as an example the case where you get ten consecutive throws of a game cube (which are all the throws made in this game) the number "1". If it is a standard die, then even though it is a rare and unusual event, we have no justification to believe that we will also get a "1" in the next roll.

However, if we have no information about the dice's standards, the fact that the chance of ten identical consecutive results is so small should arouse our suspicion that it is not a standard dice. In fact, it would be completely rational to believe under these circumstances that this is an abnormal die that shows a "1" on every roll, and therefore it would be justified to believe that the next roll would yield the same result. We can, therefore, justify an inductive inference as follows:

Assumption 1: All previous instances of A were of B character.
Assumption 2: The likelihood that this result is a coincidence is extremely low.
Intermediate conclusion: there is a reason why every case of A is of the nature of B.
Conclusion: the next case of A will also be of B nature.

The fact that I get wet every time I walk in the rain has a reason, and therefore it would be justified to believe that the next time I do it I will get wet too; There is a reason for the fact that lightning is always accompanied by thunder (the increase in pressure and temperature caused by lightning leads to a rapid expansion of air), and therefore it is justified that I expect thunder even after the next lightning; The fact that all the swans I have seen so far have been white surely has a reason (genetic, obviously), and therefore it would be justified to believe that the next swan I see will also be white (since this belief is false, it is not in the sense of knowledge, but it is still justified).

Knowing the cause is not necessary - it is justified to assume that the uniform nature of a certain phenomenon has a cause, and therefore it is justified to assume that this cause will preserve the uniformity of the phenomenon in the future as well.

It must be assumed that Hume himself would not have accepted this solution, and would have asked what guarantees us that the same reason that caused the uniform nature of the phenomenon in the past will operate in the future as well? There is no logical necessity that the laws of nature will not change over time, and from the fact that they have not changed since Galileo Galilei began the tradition of experimental science at the beginning of the 17th century, we can conclude that they will not change in the future either through inductive inference, and therefore we have not really escaped the problem of circularity in justifying inductive inferences...

Dr. Marius Cohen teaches philosophy at Ben-Gurion University.
The full article was published in the November issue of "Galileo" magazine