Comprehensive coverage

# The stacking paradox

Can removing one grain of sand from a sand pile make it not a pile? If not - how is it possible that if we persist in removing grain after grain from it, even though none of them cause actual change, we will still remain in the end without guile?

Marius Cohen "Galileo" magazine

The paradox

Similar to Zeno's paradoxes, the deception paradox is also an ancient paradox, first presented in the 4th century BC by Eubulides of Miletus, who was a contemporary of Aristotle. This paradox raises the question of how a series of small and imperceptible changes can add up to a large and significant change.

So, for example, if a pile of sand is placed in front of us, is it possible that after removing just one grain of sand from it, the collection of grains will cease to be a pile? This seems improbable, since one grain of sand is not capable of making a heap not a heap (and vice versa: it is unlikely that the addition of one grain of sand to a collection of grains, which is not large enough to be called a heap, will suddenly turn it into a heap).

However, if after removing one grain of sand from a pile of sand, the remaining collection of grains still constitutes a pile, then even removing another grain from it will not change its status as a pile, and so will the removal of the next grain, and so on. But when we are left with a very small number of grains of sand (say individual grains, or even a single grain), we certainly do not want to call what remains a heap. How is it possible, then, that removing the grains of sand from the pile made it not a pile if the change did not actually occur in any of the steps?

Variations on the paradox

This paradox can be applied to any vague concept, that is, one whose limits of application are not sharp. The term "heap", for example, is a vague concept, because there is no defined number of grains, from which onwards a given collection of grains is considered a heap. So is the term "baldness": there is no exact amount of hair, which if a person's head does not boast of it, he is considered bald. Therefore, the paradox can also be applied to this concept: if we pluck a hair from the head of a person who is not bald, it is unlikely that this action will make him bald.

That is, plucking one hair from the head of a person who is not bald does not change his status as such. But if we continue the process, eventually the number of hairs on that person's head will be so small (for example, only one hair), that no one will doubt his baldness. And again: How is it possible that pulling hair after hair from a person's head turned him from completely bald to bald, if none of the hairs pulled out actually caused this change? (That's why the stacking paradox is sometimes also called the "bald paradox".)

Similarly, a child who grows up in one day does not suddenly become an adult (if a person is considered an adult from the day he is bar mitzvah, or from the day he is no longer a minor before the law, then the terms "child" and "adult" are not vague in this context, and therefore The paradox does not apply to them, but we refer here to the terms "child" and "adult" in their obscure biological context.) If so, how is it possible that multiple repetitions of this action (increasing in one day) eventually bring about a change? The child becomes an adult, although it is impossible to point to a specific day in his life when this change occurred.

The logical problem

The stack paradox can also be formulated as a chain argument, which has this structure (where each letter represents a claim):

Assumption 1: A
Assumption 2: If A then B
Assumption 3: If B then C

........................... ..

Last assumption: if Y then Z
Conclusion: Z

If each of the premises of the argument is true, then its conclusion is also true, that is, it is a valid argument. This follows from the fact that if the first two premises in the argument are true, it is possible to conclude that claim B is also true (this logical move is called modus ponens in professional parlance). If the third assumption is also true, then with the truth of statement B one can conclude that statement C is also true, and so on. The formulation of the stacking paradox as a chain argument is done as follows:

Assumption 1: A million grains constitute a stack
Assumption 2: If a million grains form a stack, then a million grains missing one also form a stack
Assumption 3: If a million missing one grain constitutes a heap, then a million missing two grains also

constitute a hoax
................

The millionth assumption: if two grains constitute a heap, then one grain also constitutes a heap
Conclusion: one grain is a cheat

The first premise in the argument is true (if you don't think so, start the argument from a larger number of grains), and all other assumptions are true based on the assumption that removing one grain from a stack cannot make it not a stack. But since all the premises in this chain argument are true, it follows that its conclusion is also true, and this is contrary to the fact that no one would call a single grain of sand a "stack". One can of course argue that the very fact that the conclusion of the argument is clearly false indicates that at least one of the premises of the argument is also false, but this means that there is a certain number n of grains of sand (or hairs). A collection of n grains of sand does constitute a pile (and a man with n hairs on his head is still not bald), a collection of n-1 grains of sand is no longer a pile (and a man with n-1 hairs on his head is already considered bald), and this seems completely improbable.

Attempts to resolve the paradox

Much has been written about this paradox, and the number of attempts made to solve it is enormous, but there is no elegant solution that is agreed upon for everything. I will present some of these attempts, and in advance I will note that like the rest, they are also controversial.

Solving the fuzzy logic: Classical logic is unable to handle fuzzy concepts. Classical logic uses only two truth values: "true" and "false". That is, a claim can be either true or false, but not accept intermediate values ​​and be, for example, "not completely true, but also not completely false." However, valid claims of vague concepts such as "cunning", "bald", "child" and many others cannot belong to this category.

We are not always sure whether a given collection of grains deserves to be called a heap; Some people are said to be not quite bald, but pretty close to it; And regarding boys in a fairly wide range of ages, we say that it is not correct to call them "children", even though they are not yet adults in the full sense of the word. If so, a person can be completely bald, a little bald, not so bald and not bald at all, and therefore the claim "this person is bald" may not be true but also not false, and it is appropriate to allow it to have intermediate values ​​between true and false.

Such logic, which makes use of different degrees of truth, is called fuzzy logic, and those who turn to it to solve the paradox will say that the claim "n grains constitute a pile" is not always either true or false, but has a degree of truth that depends on n, that is, on the number of grains in the collection ( The greater the number of grains, the higher the degree of truth of the claim). Hence, each of the assumptions in the chain argument that we presented above also has a different degree of truth.

The proposition "If a million grains constitute a heap, then a million grains lacking one constitute a heap" does indeed have a high degree of truth, but the following propositions have progressively smaller degrees of truth (and depend on the truth values ​​of its components), and therefore the argument as a whole does not lead to the conclusion that even one grain constitutes a ruse. If so, the transition from "deception" to "non-deception" does not happen all at once, but gradually, and the paradox only arises when one tries to apply (and improperly) classical logic to vague concepts.

Not all the assumptions in the chain argument are true: there are collections of grains that we would have a hard time deciding whether they deserve to be included in the concept of "stack" (or people that we would have a hard time deciding about whether they are bald or not). However, if we are pressed to choose between the two alternatives: "deceit" or "non-deceit" ("bald" or "non-bald"), then there is a certain number of grains (of hairs), for which it is determined, albeit with great hesitation, that the collection is a hoax (the person is not bald). After removing one grain from the collection (pulling out one hair), we are forced to determine, again with great hesitation, that what is left in front of us is no longer a pile (this person is now bald).

Although, this specific number of grains (hairs) may vary from person to person and even from trial to trial for that person, but the transition from a pile without a pile (full-bald to bald) does happen in one of the stages of the process. The fact that it seems improbable to us that the removal of one grain (pulling out one hair) could cause a change in our perception of the validity of the concept is due to the fact that, from the enormous number of possibilities, it is difficult for us to imagine the exact number of grains (hairs) that would lead us to a change in perception, especially in light of the fact that this change It is so tiny that our determination is accompanied by great hesitation.

Explanation of the two components: When a sand pile is removed grain by grain, and we are required at each step to determine whether the remaining collection of grains is still a pile, we do so on the basis of two cognitive components: one component is comparing the remaining collection of grains to our subjective perception of the concept of "heap", and the component The other is comparing this collection to its state in the previous stage, before the last grain was removed from it. The assumption "if n grains form a stack then n-1 grains also form a stack", is true with respect to the second component only. That is, the chain argument is valid because each of its premises is true in relation to the second cognitive component: the change from stage to stage is so tiny that we are not able to notice it.

However, since our actual determination is also affected by the first cognitive component (comparing the state of the object to our subjective perception of the concept), it is possible that when the number of grains reaches a certain amount, the term "heap" is determined in relation to our perception, because the collection of grains is no longer a "heap".

## Comments

1. Dodan Responded:

Obviously, this is a matter of definition, and definition is a matter of cultural necessity. When is "soft ice" soft "ice" and when is it "snow"? We do not discriminate. The Eskimos, for whom ice is an essential building material, have nine different names for the types of snow and ice. If it were essential for us to define different "deception" situations, the paradox - which is a product of the ambiguity of language - would not exist
And in the same matter - is the ice that combs his hair using the "savings and loan" method - from the back of his neck towards the forehead, is it ice or not?

2. Kelly Responded:

If we flatten a million grains of sand they will not be called a pile
If we take ten large stones and arrange them in a pyramid shape it will be called a pile
Hence a stack is a kind of pyramid composed of asymmetric planar layers

3. by Michael R. (formerly Michael) Responded:

Noam:
The one who first introduced this as a definition problem is me (see response 4 and on).
I still think that's exactly the problem.
All definitions in our world are arbitrary and it is even possible to argue whether it is justified to call a mug a "mug".
The logic only works on defined terms and if you want to use it - there is no escape from definitions.
When it comes to applying logic to terms that have not been precisely defined - there is still room for solving the problem through a precise definition based on statistics (for example by asking many people what is the minimum number of grains they would consider a pile and adopting the median as a definition. This, of course, solves the question of justification you raised) .

4. The anonymous user is me…

5. Anonymous (unidentified) user Responded:

Isaiah the servant of God

It seems to me that it is not so simple to avoid the pardanx.
Let's say for the sake of it, that we define a stack as 10,000 grains or more. Apparently the paradox is resolved - as soon as you get down to 9999 grains - there is no pile.
But here comes a simple question:
What is the justification for defining 10000 grains as a pile, and one less grain is no longer a pile.
This seems to be a completely arbitrary definition, which is not sure to settle the matter.

6. Anonymous (unidentified) user Responded:

The paradox falls under the tyranny of the discontinuous mind.

7. Fan Responded:

Here is my opinion on the paradox, and to the inventor of the paradox - you should close your eyes:

Are they trying to confuse us? They confuse themselves! Here are the facts again:

1,000,000 grains: stack.
1,000,000-1 grains: stack.
1,000,000-2 years later: no. (From my point of view, maybe others think that minus grains is a pile..)

In short: a million? pile. -3,000? not a pile Wow, what a paradox!

8. nonsense in my life Just linguistic sophistication.
You have a pile of grains. You removed a grain, you have a smaller pile. Adding a grain, you have a bigger pile. "Heap" is not a mathematical constant.
The example of hypotheses is also out of place. If you pull out her hair the man will not be bald, if you persist in pulling out hairs you will only get beaten by the man and maybe you will be bald.

9. Isaiah the servant of God Responded:

To my astonishment, I have to agree with my ideological opponent Michael (unless we are talking about the former Michael R.) - it is clear that in order to check the existence of a condition ("is there a heap") I must pre-define the condition, at least at the level of a range of values. Otherwise, what we have is not an issue of logic, but simply a social experiment - what is the smallest amount of grains that a random sample of people would be willing to define as a "pile". But, if you allow me to try to open a new direction here for a moment, I wanted to ask - why not continue? Well, if one grain meets the condition (we'll ignore the lack of definition for a moment), why not half a grain? And if half a grain, why not a quarter, etc. And in short, we will accept that the minimum size of a pile tends to zero. In this respect it is perhaps similar to Zeno's classical paradox.

10. But wait, wait, wait... it looks like he got stuck on the small things, it all depends on how the person defines a stack... if a stack according to a dictionary or an encyclopedia, it means that more than one object that is placed in a certain proximity to each other is already a stack, so yes, put two grains of sand and you have a stack, so one You no longer have one, add one again, you have, and even if this is not the definition, there is still a certain threshold that the same individual who piles the grains on top of each other will decide that this is a trick... are you entering the language?! for expression? For understanding?... because everything they say here is a figure and every step forward will not change the figure, but again it all depends on the definition, just a retarded example... thank you, focus for a moment on the whole word thank, go deeper but really in the sound of the meaning and stop and think for a moment... what the hell is this, who farted the sequence these sounds and decided that it means a positive expression and then if you want more you can go even deeper two people their whole lives were together they learned everything together everything is the same one is a color blind who sees black and white and the other doesn't now between all the colors in the world there are differences in texture and strength ZA Pink in black and white won't look like yellow.. a certain discernment is needed but he had his whole life to practice it in ignorance of course now think for a moment me and him are looking together at a color we both know what the color is called but who said this is what we see?!… Who said it's not green?..or white?...or red??....in the end all the paradoxes that lack mathematical and physical foundation are just a distortion of what is known in advance that you try to delve into something that is not meant for delving into and then you find the holes in the system and you simply perceive You are a human being, no more and no less than a bald monkey, so please leave you with such nonsense and go learn more necessary things..for all of us

11. Michael Responded:

Jacob:
You are just being arrogant and missing the point.
We will see you define "red" in the definition of children.

12. Jacob Responded:

Philosophy in the divorced man's room. And why?

Grain-called grain.
2 grains-are called arama.

When we were little, we used to play abortions and then when children would fall on top of each other, we would call it a 'pile of children', it was at least two children lying on the floor.
And I always claimed that wisdom should be learned from children..the adults only complicate matters.
Jacob

13. Michael Responded:

shocked:
I think you're wrong.
I know dumb logic has bought it a foothold in our circles but I still think it's just probability for the lazy.
When you postpone dealing with reality, all you achieve is an increase in arbitrariness in the final decision.
There is no problem defining "red" and I assure you that any such definition (which will not be based on wavelengths because "red" is neither a wavelength nor a range of wavelengths - but, for example, on relationships between excitation levels of three standard gauges), will serve us no Less good than any vague definition.
I do not claim that it is always correct to be diligent and give probabilistic definitions.
Sometimes it is not worth the effort and it is better to use "probability for idlers", but in principle it is clear that probabilistic definitions will work better in the end.
The argument with the rational numbers is really strange because it does not single out probabilistic calculations at all.
Is numerical analysis that approximates the behavior of physical systems also forbidden, in your opinion, to be used?

14. shocked Responded:

Fuzzy logic uses elements from Evidence Theory and because of this it includes the theory of probability. Using an exact definition and division functions inevitably causes calculation errors (like using rational numbers in calculations because there is no such thing as an irrational number). To all those who announced that the stacking problem can be well defined, how would you define, for example, the color red? (A single wavelength? A range of wavelengths? Does the human visual system have the ability to distinguish between light frequencies with such a good resolution?) And despite the dim definition, we use the color definitions with almost no special problems. Any probabilistic model will include missing or incorrect information and therefore necessarily an error. Fuzzy logic makes it possible to ignore the error until the last calculation where we must return to the real world.

15. Semyon Ristov Responded:

A somewhat problematic intermediate situation can arise:
Let's say there are 9 grains and they are a pile. 6 grains is a "hesitant stack" and 4 grains is a "hesitant non-stack". And what about 5 grains? Are they "both heap and non-heap at the same time", or are they "neither heap nor non-heap"? Or are they even Schrödinger's cat?

Have a nice day.

16. Anonymous (unidentified) user Responded:

Girl, you are bored. You can use the time for other nonsense.

17. Michael Responded:

And here comes infinity and demonstrates how accurate my claim was that everything here is actually a matter of definition.
And for the smart farmers - I suggest you read what I wrote on the subject of obscure logic.
I think I'm the only one who even talked about it and I didn't claim that it has no applications. I only said, and I repeat, that wherever vague logic is used it is only because one is too lazy to translate the problem into probabilistic terms.
In the end, I estimate that laziness pays off because while probability has clear laws on the question of how, for example, to calculate the probability of a grouping of a number of events, in vague logic there is only ambiguity in this matter.

18. infinite Responded:

Even one grain is a pile, a pile of molecules which are also piles of atoms which are also a pile of subatomic particles which are also a pile of vibrating strings of energy which are also a pile of something that science cannot yet know and so on ad infinitum

19. ב Responded:

A note to dog owners, you should always remember to pick up the pile

20. ב Responded:

This is a convergence problem. It's a paradox only if you choose to get stuck in endless chain arguments. As we know, any column that we know converges even if it is infinite, the speed of its convergence depends on the pattern of the column. Usually adding more dimensions makes it easy to see the way of convergence. such as pie. If we treat each of the digits as a seed from the stack then there is no end. But pie is finite just like the stack remains to be seen how it shrinks.
Pi can be written in at least 20 different series that converge, most of them are derivatives of the zeta function.
Clearly the chain argument is not infinite if it is actually a copy of one member or a subset of one of the columns. It remains to check whether all these series form isomorphic groups and whether it is possible to define a group of paradoxes of this type and check whether they constitute a subgroup.
The stack itself can be divided into stacks in binary order so that we get a very small stack very quickly. And you can also go in the opposite direction, that is, take a grain from one pile and place it on a clean surface. and see if we will die of boredom before a heap is actually formed into a sufficient collection that we can call a heap.
I guess these articles are specifically for cucumber season.
In the next article you will check when the color dries when it is printed N times on the same page. And then the column goes down or up as you wish.

21. A smart farmer Responded:

You miss the point, this is not a philosophical question. Fuzzy logic has countless applications in decision-making systems, artificial intelligence, and more... Considering that in real life there is no black and white, everything is gray, many functions have been developed to define such "gray boundaries".

22. A. Ben-Ner Responded:

A similar paradox is the "naked" paradox.
Here is the "paradox":
A naked girl, according to the definition I chose, is a girl whose whole body is bare, without covering. Now suppose that a naked girl wears a wedding ring on her finger. Now her whole body is no longer bare, the ring covers a small part of her body. Would she still be considered naked?... .Depends on who you ask, and what his definition of "naked" is.
Of course, you can continue to detail the example for many other items of clothing such as: earrings, socks, hats, glasses, hats and much more...
The answer to the question of whether the girl is still naked does not depend, therefore, on the specification of the coverings covering the woman's body, but on the definition that each and every one gives to the concept of "naked".

23. A. Ben-Ner Responded:

In my opinion, the "heap paradox" is neither a logical nor a mathematical paradox. This is a pseudo-paradox, i.e. a simulated paradox, which is based
On a distortion in the understanding of the word that bears the name paradox. In the case in question - the meaning of the word "heap". That is, it is "as if a paradox" arising from a mismatch between the meaning, defined in the language, of the concept and the associative meaning that the word evokes in the minds of many of the people who hear or read it.
In the case before us, the real interpretation comes from the root of the word - "heap". i.e. some object
The term on another object that is under it. For example, the chair can be stacked on the table.
Hence a stack consists of two or more objects. And that is the solution to the pseudo-paradox.
However, in the associative context, most people see, when thinking about the word "stack"
From a small hill that includes many small components, which are difficult to count, such as a pile of gravel or a pile of sand and this is the source of the paradox.

24. Sabdarmish Yehuda Responded:

It is clear that first of all we need to define the concept in question (in our case - trickery) and only then check if a change we made still leaves us within the scope of the definition.

Let's say I define a group of grains from 10 grains or more as a "stack of grains". It is clear to me that if after a change I made in the pile, if I still had 10 grains or more left then I was left with a pile of grains but if after the change I was left with fewer grains then I was left with no pile. It all depends on how I defined the term:- "heap of grains".
And just for good measure, in the case of infinite stacks the situation is a little different.
Suppose I define an infinite set of prime numbers as a "stack of numbers".
We will now take the set of all natural numbers, after all according to our definitions it is an infinite set of numbers and therefore it is a "stack of numbers" now every time we remove a natural number from this stack we will still be left with a stack. And even if we subtract an infinite set from the stack of natural numbers, we may be left with a stack, for example, if we remove all even numbers from the natural numbers, we will still be left with a "stack of numbers".

good afternoon
Sabdarmish Yehuda

25. Michael Responded:

Eran:
You adopt my words exactly and provide your own definition of the term deceit.
You must agree with the fact that since the definition you used does not appear in the dictionary there will be people who will claim that even one grain is a pile and there will be people who will claim that even 10 grains is not a pile.
Everyone has the right to define the stack as they wish.
The only thing he has no right to do is not define and expect defined results.

26. Eran Responded:

There is no paradox here, I don't understand why a whole philosophy is made of this nonsense!

One grain is not a pile.

2 grains make up the smallest possible pile.

3 grains is a slightly bigger pile.

4 grains is an even bigger pile…..

And so on and on….. There is no paradox here just a pile of nonsense.

27. Michael Responded:

Offhand remark:
The author, by the way, insisted on the fact that the problem stems from the vagueness of the definition but insisted on trying to apply the laws of logic to it anyway and as if he was surprised that the result was vague.
It's like we take the addition operation which is defined and should give exact results and we complain that we don't know how many are 5+X where X is the number of hairs that were on Olmert's head on Waldo day.

28. Michael Responded:

Download software:
You didn't understand my words.
When I talked about a specific stacking paradox I was talking about the fact that the problem stems from not defining the term stacking precisely. A precise definition of the term (say 712 grains or more) will raise the paradox.
I talked about the subject of statistics when I explained what I think is wrong with the vague logic.
You just made a salad out of things.

29. Life Responded:

What about the cumulative effect of turning a stack into a non-stack?

30. It has nothing to do with statistics.
The question is what do you as a specific person define as deception, and when exactly is the transition to "no deception" considered. The writer, in my opinion, gave a good answer, which I also thought about.
Best regards
יוני

31. Michael Responded:

In my opinion, this is not a real paradox, but one that arises out of laziness.
I'm too lazy to define exactly how many elements there are in the pile and then shout Weloti when in the absence of a definition we can't draw conclusions.
I'm not saying that you must define a stack precisely - I'm just saying that you shouldn't expect imprecise definitions to enable precise logic.
To me, by the way, this whole thing of fuzzy logic is simply probability for the poor or - again - more precisely - probability for the lazy.
When there is a word that is not precisely defined, there are still statistics that can be done and find out what is the probability that a person will say about a collection of elements of a certain size that it is a pile and then say that it is stacked in a pile with probability X.
You are too lazy to do the statistics and instead start talking about vague logic.

32. דני Responded:

I don't know, it just seems to me like a proof by induction that doesn't work because they forgot to add the base..

33. point Responded:

The term "pile", like the other terms, indicates a mindset, and therefore has no logical meaning at all.
That's why this whole paradox has nothing to do with logic but with the humanities (or society).

34. Ami Bachar Responded:

fun to read Differential logic.

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