**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

## 22 תגובות

The main thing I learned from Marius and deductively is that he is a great teacher

point:

I don't know what you are talking about.

Elijah:

You did not understand me.

When you talk about necessary conclusions - you conclude their necessity based on the laws of logic.

And how do you deduce the laws of logic?

You do not deduce them - you always assume them as axioms (and add these axioms to any other axiomatic system such as the axioms of geometry).

By what right do you do that?

Simply because you cannot do otherwise because evolution has embedded these axioms in you and you cannot think without them at all.

And why did evolution do this?

It is - as I said - an inductive process in which evolution made experiments with all kinds of organisms whose logic works in a way that contradicts our logic, these experiments failed (the organisms became extinct) and only organisms remained whose inherent laws of logic work normally.

In other words - evolution here also "drew" conclusions through the experiment (the results of which were not self-evident to it, if only because of the fact that it does not understand anything) and ultimately instilled in us a sense of necessity regarding certain laws of logic - not because they are really necessary but because in all the attempts it made – They worked and others failed.

Michael,

Deduction moves from basic premises to their necessary conclusions. It is not necessary that the basic assumptions are correct at all, but nevertheless it is a logical-analytical operation.

Induction is an intuitive - synthetic jump, from the particulars to the underlying rules. Although the basic assumptions are not proven and we only have an intuition that they are true, I do not see how deduction is an extreme case of induction. The evolutionary explanation does not help us at all.

point,

Yes, there is a problem, and on the other hand, it's a complete waste.

Michael, there is no induction problem. There is a problem with those who think there is an induction problem.

Elijah:

The axiom of parallels is not necessarily true and the whole basis of general relativity is that in the real world it is not true.

As I said in response 6 - the situation is much more extreme and everything we call "deductive" is actually inductive which is established by evolution and determined in our minds.

Eliyahu and I created:

Descartes' claim is also not an exception and is by definition a conclusion based on mathematical logic.

It is true that our logic existed even before mathematics, but this is irrelevant - today - after taking it and adding all kinds of conclusions arising from it in the form of mathematical sentences - it is customary to call the whole complex by the name "mathematics".

creative,

Every deductive claim is built on basic assumptions, which themselves cannot be proven. For example, Euclid's geometry, which is based, among other things, on the axiom of parallels, which no one has yet been able to prove. And yet we believe that they are true.

You are certainly right about Descartes, but note that from there it was very difficult for him to advance and prove the existence of the external world to him. In the next step he tried to prove the existence of God (at least according to Dr. Yuval Shtinitz in his book) and the proof convinces only the convinced (like me...) - in my opinion it is based on unnecessary assumptions.

There is a wonderful book (hard to read but worth every word) by Rabbi Dr. Michael Avraham: "Two Carts and a Hot Air Balloon". The book develops the whole topic of raised in the article.

Excellent article! Well done.

What to do not everyone is able to understand philosophy and therefore talk nonsense like a point and an era.

It has nothing to do with mathematical induction - which in general is a deductive inference method in number theory that gives inductive power.

Elijah:

If I understood correctly, you said that it is impossible to draw a necessarily correct deductive conclusion outside of mathematics. But I would say that Descartes' cogito ergo sum is a deductive proof of the existence of the self.

Marius Cohen uses an idea put forward to him by Taleb Nissim Nicholas who wrote the books Black Swan and Illusions of Randomness and does not even mention it a word. Not pretty. The allegory of a black swan, the concept exists way back - but in the context of breaking statistics, it was done by Taleb Nissim Nichols and deserves at least a mention of his name.

point:

Not true!

The article explains exactly what he calls induction and it is not mathematical induction just as it is not electrical induction.

Eliyahu, induction is a mathematical concept and has nothing to do with epistemology and how the brain/mind works. After you understand this, you will understand that there is a place for the scientific claims I made that are not inductive but assume that the reader knows the concepts in Hebrew.

point:

I didn't understand anything and I probably wasn't convinced of anything.

Like the question: maybe KDHA will reverse tomorrow and we will all fly?

And those who believe in coincidences should ask the question will it arise?

Idan,

Your belief that when you move the mouse the cursor will also move is based on induction.

Why won't the sun go out tomorrow morning? Because you haven't seen other stars do that. So how do you project from other stars to our sun? - Induction.

Every inference we make from the particular to the general is induction, and in fact, deductive thinking only exists in mathematics.

point,

Your scientific explanations will not answer the problem of epistemology because you are already starting from the premise that education is rational and usually correct.

It's a shame that there isn't an extension of Yom's philosophy here and from it to Kant - both form the basis of the philosophy of science.

And the goal is very simple. Methods that use the word convention and imagine that they describe physical facts by statements in language, are prone to internal contradictions, which should be avoided.

Michael My opinion is based on what we know about neural networks and how they learn. The language does not describe the processes that lead to the creation of the language. The same goes for mathematics. Yom's answer of "Hagel" is the most accurate. Because what actually happens is that the network is shaped by certain weights, etc.

point:

"Covenant" is a legitimate word in this context and it is the action performed by those who believe.

This is not a new word and I think it has a place in the description given by the author.

The author also explained what he calls "induction" and it is clear from the words that this is not a mathematical induction but a certain type of inference - "whose premises lead to his conclusion with high probability, but not absolute certainty, and is characterized by generalization based on known private cases".

Beyond that - there is no need for a child to know what induction is in order to perform it - just as you are able to be aware without knowing what consciousness is (like the man who one day discovered that all these years he was actually a prose writer).

Rather, starting with the belief that "everything is true" is impossible - neither as a child nor as an adult, if only because of the fact that we do not know what "everything" is.

In fact, a large part of your responses are simply a pointless rebellion against the language (which you do not hesitate to use in your formulation) and against the claim that you know anything at all (and the rebellion against this claim is also said with the assertiveness of those who supposedly "know"....)

It's hard for me to understand your purpose in this whole thing.

As a matter of fact - in my opinion it is important to understand that deduction is actually an extreme case of induction because the same rules of logic that we use to derive the necessary consequence of the conclusion are rules whose only justification is inductive.

Although it is not usually an induction that the individual performed but one that evolution "performed" and as a result instilled in the individual the absolute belief in the correctness of these rules.

The author just invents concepts and mixes many unrelated concepts.

Why say "the treaty" when you can use the word "faith" much more clearly and precisely. We believe that…

There is no connection between induction and our way of thinking. A child does not even know what induction is. In our childhood we start from the fact that everything is true and slowly find legality by negating more private laws and including the new facts.

Although it is common to think so, the logic of language has nothing to do with factual reality. "The sun will rise tomorrow" is a sentence that conveys nothing in reality, only in our fevered minds.

Idan

If you have already entered a serious website and read a serious article, then why are you embarrassing yourself and reacting in a shallow way?

What does the setting of the sun or what the religious pray in the morning and in the evening have to do with an article that deals with human problems??? I would appreciate it if you could explain

To my dear father Blizovsky.

Can we please make the ads hiding the article transferable. This way we will double space, we will also look at the ad, while it is moved from place to place on the screen space, and we will also be able to read the article in its entirety.

Thank you for your answer.

Reminds me of the religious who pray in the evening for the sun to rise and in the morning for it to set...

or whatever it is :)

Kind of a useless rain dance

Chutzmaz that the net cannot go out like that in one day like a lamp... there is a process of decay that will take a long time

Another thing I don't understand from this article

Since when is induction correct for our everyday life

We are not robots and induction works mainly on mathematics which has its own laws that do not change and can be observed and calculated in advance

And one more thing

The measurement according to time periods of 24 hours is wrong, you have to look at time intervals of hundreds of thousands of years or millions of years

Just like you would say that a car's engine has traveled a kilometer and another kilometer and even at 30000 kilometers it travels, so it will be able to travel forever,

But the chances that it will reach a hundred thousand kilometers without turning off are zero

I got confused

Will the sun rise tomorrow??? 🙁