Talk:Sorites paradox

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There is a glaring omission:

Premise 3: A heap is a large collection of grains of sand

Without this third premise, Premise 2 is only applicable once (is a large collection of grains of sand minus one grain still a large collection of grains?)

  • We start with a large collection of grains, and Premise 1 concludes it is a heap.
  • We remove a grain of sand, and Premise 2 concludes that it is still a heap.
  • Premise 1 and 2 are no longer applicable, unless we can conclude that a heap is somehow related to a large collection of grains of sand.

Of course, surely we can just do with one premise?

Premise 1: A large collection of grains of sand minus one grain is still a large collection of grains of sand.

Suggested change
with at least one grain atop at least one other grain. This eliminates the paradox; two grains side by side are not a heap, but one grain on top of another is a heap. This can be tweaked to accommodate the case where a flat surface of one grain touches the edge of another grain. Two grains can make a heap, one grain cannot. — Preceding unsigned comment added by (talk) 22:53, 8 September 2015 (UTC)
This is all original research and thus is irrelevant. (Aside from being confused and wrong. Notably, the "suggested change" misconstrues the whole issue, which is about *indefinite* descriptions. Offering a definite description of a heap of sand does nothing to resolve the paradox, which still applies to beards, baldness, and whether the first chicken came before the first chicken egg, for example.) — Preceding unsigned comment added by (talk) 09:41, 22 June 2019 (UTC)

Suggested Definition[edit]

A suggested definition for a heap: a combination of more than 2 objects that raise at least one of their members above the others. Thus four carefully arranged grains of sand of equal size could be heap (3 forming a tripod for a 4th - or two large grains supporting a third). Whereas a million grains of sand, with none on top of one another does not equal a heap.
~ender 2003-09-12 06:14:MST

For me, "heapness" is not just about "raising up"; I agreee with the Greeks that it has something to do with how many objects are involved (or, if not that, how large the collection of objects is). I do agree that a million grains none on top of any others isn't a heap, though, so perhaps we should mention raising up in the article somehow. --Ryguasu 14:28, 12 Sep 2003 (UTC)
A suggested definition for a heap -- This page is for improving the article, not for people to try to solve the paradox -- that is original research and is invariably confused and inept. (This "suggestion" misses the whole point, which is that the paradox applies to indefinite descriptions. A definite description of "heap" isn't subject to the paradox, but that does nothing to solve the problem, which still applies to beards and so on.) -- (talk) 09:57, 22 June 2019 (UTC)

The sorites isn't really about 'heaps' as such... but there is one philosopher -- I forget who just now -- who has half-seriously suggested that four is the least number of grains that can make a heap, just as ender described above.

This is part of the tradition of resolving the paradox by denying the 'tolerance' premise -- ie that there are no two elements a, b of a sorites series such that p(a) but not p(b). In other words, one resolution is to say that the definition of the predicate can be sharpened so that there is a definite cutover point. In this case, removing one grain from a heap of size 4 creates a non-heap, and so one of the steps in the paradoxical argument now fails. In other situations, however, this can be powerfully counterintuitive, IMHO.

Ornette 16:58, 3 October 2005 (UTC)

Yes. The paradox isn't about heaps specifically, but about imprecise definitions. Is a man with a single hair on his head bald? Is a man with two, three... thousand, ten thousands hairs bald? And so on, there are many definitions.
However, I think that we should mention this "solution" of the paradox in the article (while of course stating that that does not impede the paradox). Nikola 21:04, 17 October 2005 (UTC)
No, it is not the role of editors to "mention" anything other than what is found in reliable sources. -- (talk) 09:57, 22 June 2019 (UTC)

A case for name change?[edit]

I suggest that this article should instead be called The sorites paradox, and that Paradox of the heap should redirect here, rather than the other way around. In the philosophical literature in which this paradox is discussed, it is, I think, more commonly referred to as "the sorites paradox" than as "the paradox of the heap". Such a name change would be less prone to suggest that the philosophical problem which this article is about is a problem specific to heaps. Opinions? Matt 9 Nov. 2005

I tend to agree. Ornette 18:02, 8 November 2005 (UTC)
I agree as well. --Pfafrich 21:04, 8 January 2006 (UTC)
I definetely agree, change the name to Sorities Paradox, it is better known that way. 01:43, 19 April 2006 (UTC)
Well, maybe not that well known, because you misspelled it. —Keenan Pepper 02:24, 19 April 2006 (UTC)
I strongly agree. Maelin 10:31, 31 July 2006 (UTC)

As long as whichever one redirects to the other, it doesn't matter, both names are used in the literature. Yesterdog 00:47, 16 May 2006 (UTC)

Case for a merger?[edit]

There are four articles in Wikipedia dealing with essentially one and the same philosophical topic: Imprecise language, Paradox of the heap, Vagueness and Continuum fallacy. (Sorites paradox redirects to Paradox of the heap.) I have done a little editing of the Vagueness page, but really I think all four pages should be merged, or that at very least, they be rationalised to two pages, one a longer one on the philosophical problem of vagueness, and the other a quick summary of the sorites paradox with a link to the vagueness page for a more in-depth discussion. What do people think? Matt 9 Nov. 2005

I'm not sure that I agree. These are related, but not neccessarily the same topic. The fallacy which uses the paradox is something different from the paradox. (And, add to the list the Ship of Theseus.) What benefit would there be from a merge? Nikola 19:09, 9 November 2005 (UTC)
Sure. So I now have a more modest proposal (See Talk: Imprecise language.) Btw I think Ship of Theseus is certainly a different thing. User:Matt9090 10 Nov. 2005
This is confused. The whole point of a logical paradox is that apparently valid inferences result in an invalid argument, but it's not clear what's in error -- it's not possible to separate "the fallacy" from "the paradox". -- (talk) 09:57, 22 June 2019 (UTC)

Removed section[edit]

The section entitled "idiot's solution" did not make a lot of sense. I could not discern what the "solution" actually was that was being offered. Also note that Zeno's paradox of the frog jumping across the pond in jumps each 1/2 as big as the last etc. has very little to do with the paradox of the heap. They are different puzzles with different solutions. The new section seemed to suggest there was some sort of connection but left the reader with no idea what this connection actually consisted in. So I deleted the new section. User:Matt9090 14 Dec. 2005

Differences in Ages[edit]

This paradox reminded me of something I've heard often. It's often said that age x is essentially the same as age x±1 (ie. "What's the difference between 25 and 26?" as an argument for still doing something done at 25 at 26), which infers that the difference between x and x±1 (in terms of ages) is negligible. Obviously, this can be extended in both directions infinitely: by applying this principal recursively, the difference between any number and any other number can be considered negligible (in the case of ages, at least). Logically, of course, this doesn't make sense. I'm not sure if this is an appropriate concept to discuss in this article. --Dvandersluis 20:41, 23 June 2006 (UTC)

Actually, I think this argument isn't very well-liked among philosophers. If you can find a lot who argue for it, feel free to put it up, but keep in mind one argument: 25 and 26 aren't essentially the same, they merely are so close that they're almost the same thing (much like how 1.1 rounds to 1.) However, 25 and 40 are more equivalent to 1 and 20:50, 24 November 2006 (UTC)

Three Valued Logic[edit]

The problem of describing resolutions to paradoxes is that one has to understand why something is a paradox to begin with and why a paradox is resolved by your solution.

From the page:

Three valued systems do not resolve the paradox as there is still a dividing line between heap and unsure and also between unsure and not-heap.

How is this not a resolution? We've resolved it by defining set boundaries.... it may not be a satisfactory resolution, but it would appear to give a simple though arbitrary answer. It is as much a resolution as the Setting a fixed boundary solution. It would seem there should be better phrasing or wording here as to what is meant. Maybe a sentence about how the the valued logic solution no better matches our intuition than the aforementioned solution, or something.

Root4(one) 04:16, 4 December 2006 (UTC)

There is no paradox[edit]

This isn't a paradox at all. The so-called paradox is just an obscure look at the human mind's ability to establish abstract entities and associate them with words. A "heap" is a word used to describe one's relative perception of any arbitrary quantity of particles which resemble a familiar shape formed by gravity. You may as well ask how unattractive someone has to be in order to be considered ugly: obviously this is relative to the person perceiving.

The real problem comes from trying to define a heap by the number of particles in it, which has absolutely nothing to do with a heap. Who ever said anything about particle quantity? Why is that somehow implicit? The laptop I'm writing this on is not a laptop because of the number of buttons it has, it is considered a laptop because of its size, look, and feel. Trying to define something with irrelevant premises is obviously, always going to be impossible. It's like trying to measure the volume of a swimming pool with a yard stick.

With that said, it's not technically impossible to define a heap, it's just impractical. We don't know our brains well enough to examine the information they harbor. If we did, then we could determine a group of people's thresholds for what qualifies as a heap, which would consist of an abstract concept of its shape, volume, material, etc..., having varying degree of certainty between samples, since some people have obviously seen more heaps in their day and can better identify them. This could all be averaged to come up with a technical definition of the requirements that something must have in order to be classified as a heap. But what's important to realize is that those requirements would be massive with ranging sizes, shapes, colors, textures, and senses; not a simple range of particle quantities. If you must insist upon looking at it mathematically (numerically? whatever), then you can compare it to rounding a huge amount of data that we don't know how to interpret (mental abstraction) into a very simple integral form that we can interpret (particle quantity), and then trying to tell the difference between these integers after the conversion.--RITZ 15:16, 27 January 2007 (UTC)

And hence you would prove the argument made by Unger, and some others, that because the heap is a concept of the mind, composite objects do not exist (mereological nihilism) because composite objects (e.g. the chair I am sitting on now) are "heaps" of smaller things. Our misconception of what defines a heap attests to the fact that we have a badly distorted view of what reality actually is.
There are greater concerns in addressing this paradox (at least for the materialist) than deciding whether it is actually a paradox or not. In my opinion, I agree that this is not a paradox. Just like Zeno's paradoxes are not paradoxes because they go against the assumption of objective reality, this is not a paradox because it does not go against any other favorable conclusion. Mereological composition is unexplainable and is largely an assumption based upon common sense perception. --Shotgun_method 14:28, 22 Jan 2008 (UTC)

Minor Error[edit]

I think there is an error in the following quoted section of the article.

"On the face of it, there are three ways to avoid this conclusion. One may object to the first premise by denying that a large collection of grains makes a heap (or more generally, by denying that there are heaps). One may object to the second premise by stating that it is not true for all collections of grains that removing one grain from it still makes a heap. Or one may reject the conclusion by insisting that a heap of sand can be composed of just one grain."

The final method of rejecting the conclusion (by insisting that a heap of sand is composed of just one grain) is in fact an affirmation of the conclusion is it not? I've never read any philosophy at all, so I'm not editing this, but I assume this was just an oversight.

The argument here was that a position is possible in which one argues that a heap of sand can be consisted by even a single grain. in other words it is saying that even the last grain left is a heap in itself.
besides one could also go further by making a heap the synonym for a set and then it can be argued that zero grains left is just an empty set of sand, eg.: an empty heap. and yeah, descriptive and prescriptive approaches can be contrasted. i think the original (linguistically based) paradox arises from pretending to have a fixed definition of what exacly makes a heap, while in reality it is just a stereotype like all words of the natural language (like table, or dog). so the take away from this paradox seems to be that it calls attention to the natural language's undefined nature, which is usually forgotten by it's users.
also the original "paradox" seems to aim at the conclusion that (change in) quantity amounts to (change in) quality. (talk) 22:33, 22 May 2017 (UTC).

Resolutions dubious[edit]

Some of the 'solutions' here are a bit dubious. The 'trivial solution' is perhaps misnamed. The 'Multi-valued logic' solution is badly written - it's not clear whether it's suggesting having three predicates for categorising putative heaps ('is a heap', 'is not a heap', and 'is neither a heap nor not a heap'), or that we should distinguish between three truth-values (true, false, neither true nor false). It also leaves out fuzzy logic, which has an infinite number of truth-values. And it uses an epistemic term ('unsure') for the middle category, suggesting it is an epistemic approach. The 'visual definition' and 'group consensus' solutions need citations; I'm not sure whether they represent original research, but they certainly aren't mainstream. It's hard to see how the visual definition solution is meant to work. The group consensus solution talk of 'probability' is very strange; (if 9 out of ten people thing a particular pile of sand is a heap, the chance of it's being a heap is 0.9?) - presumably it's just a variant on a fuzzy logic approach - once we get into the vague region, the truth value of the claim that the pile of sand is a heap is determined by group consensus. We also need a section on supervaluationism and the epistemic approach. If no one has any objections, I'll start making some changes in due course. 18:37, 5 February 2007 (UTC)

The group consensus might could be worded a bit better, but I don't see what's incredibly strange about it, at least in concept. However, its a bit curious how one should test the "heapiness" of the sand pile using this definition. Are we showing the people a picture? (see what's written in "Visual Definition" for some problems with this) Putting the sand in a bag and handing it to people so they can weigh (by intuition)? Giving them a figure of its volume, or weight, and let them reason about that alone?
Otherwise, assume you have some some measure M of the pile, some test on that measure, and you have a random selection of people. Run the test by each person, and conceivably with enough tests and enough people, you may have some estimate of the percentage P(M) of people that would call a pile of measure M a heap. Since we're only concerned about what people are calling a heap (ignoring Gods, Gorillas, and Flying Invisible Pink Spaghetti Monsters), we could say we are estimating the "chance" that (a person) calls the pile a heap.
I'm fairly certain the "Visual definition" is original research. I recall one day finding it posted here and the original rendition made little sense whatsoever. It was almost like the original poster did not understand the paradox. However, I found the idea interesting, and the concept is a bit related to some research I did in the past, so I tried to fix it. My past related research was using a computer to find blood in endoscopy videos, which required a "visual definition" of a "notable patch of blood" (read this as "sand heap"). Of course, if you get into visual definitions, you get into a lot of murky areas... like what precisely identifies sand? Its color? its visual texture? And how is that measured? Is it both color and texture? In what combination? If its a linear combination, what weights should be used? Or worse, what if its not a linear combination? (HEAD BANGS WALL).
I'm sure I could eventually dig up some computer vision papers related to this topic. These papers certainly wouldn't be related to finding sand and heaps, but with some name substitutions, you find the research boils down the same problem attacked by identifying heaps by the "Visual definition" approach.
But please, if you have some ideas about making some improvements, please go right ahead. After re-reading the "Visual definition" section, I may scrap that section myself.
Root4(one) 02:00, 6 February 2007 (UTC)
Hi Root; thanks for the comments. The problem with the group consensus resolution is that it talks of probability. One could estimate the probability that a person in a given population would call a pile of sand a heap, but that doesn't immediately bear on the question of whether it is one. The resolution says that the probability of the pile being a heap is fixed by the probability that a person in a given population would call it one, and this seems crazy (it talks of the 'expected value' - I don't know enough stats to know whether this is different; I suspect one should say that it is fixed by the distribution of responses of people in a given population). Suppose the chance of a person calling the pile of sand I have in my garden a heap is 0.1; according to the resolution, there is a chance of 0.1 that when I open my curtains, I'll see a heap - that's very odd. Presumably, the thought behind the group consensus approach is that the degree to which something is a heap is determined by what proportion of the population would call it one. Or to put it another way, the degree to which the predicate 'is a heap' is true of a particular pile of sand is determined by the proportion of the population that apply it to that pile. But once you start talking of degrees of truth, you're just back to a multi-valued logic solution, with the added claim that group consensus plays a role in determining truth-value.
With respect to the visual identification solution, one again wants to keep separate the issue of whether we visually recognise a given pile of sand is as heap (and that of how we do so), from the question of whether it is one. Presumably the visual resolution wants to link these two things somehow, but doesn't give any indication as to how it should be done. It should either be sourced and clarified, or deleted.
One issue is that the article on vagueness outlines some of the possible responses much better... what are the conventions on duplicating material? 11:38, 7 February 2007 (UTC)

Flawed Example???[edit]

Consider a heap of sand from which grains are individually removed. One might construct the argument, using premises, as follows:

   * A large collection of grains of sand makes a heap. (Premise 1)
   * A large collection of grains of sand minus one grain makes a heap. (Premise 2)

Repeated applications of Premise 2 (each time starting with one less number of grains), eventually forces one to accept the conclusion that a heap may be composed by just one grain of sand.

And why is that exactly? How is one grain of sand "A large collection of grains"?


That's exactly the point. Our two apparently acceptable premises lead us, by perfectly valid reasoning, to an obviously unacceptable conclusion. We have to deal with this problem. It's not enough to say, "Oh, well, looks like we were wrong here," and then just move on. The point of the Sorites paradox is to demonstrate the property of vagueness. The word heap is not a strictly defined term. Some collections of sand definitely are heaps, and some collections of sand definitely are not heaps, but there are some collections of sand where we can't really make the call. If we defined a "heap" as being "any pile of sand containing greater than 50 grains" then the problem would disappear, since Premise 2 would not be universally true. However, we do not have such a clear definition of "heap", so we need to develop new ways of handling this problem. Maelin (Talk | Contribs) 00:34, 15 February 2007 (UTC)
I wasn't clear. I understand the issue of vagueness surrounding what exactly constitutes "a large collection of grains", I just disagree that TWO grains of sand would ever be confused for "A large collection". If Premise 2 was "A Heap of sand minus one grain makes a heap" then I would agree that I would be forced to accept that a heap may be composed of one grain, but by putting the "large collection" qualifier on Premise 2 I can't see how I could EVER be forced to accept that one grain makes a heap. Which is why I asked: How is one grain of sand "A large collection of grains"? ~dongray2
Uhm... it isn't..? What makes you think one grain does comprise "a large collection of grains of sand", or that there is any reason to assert that? Your premises above, I might note, do not force us to accept anything as being "a large collection". If we can decide what a large collection is, the premises force us to accept that whatever a large collection may be, it is also a heap, and it would also be a heap if we removed a grain of sand from it. But no matter what we decide comprises "a large collection", the premises never force us to accept anything else as a large collection. Maelin (Talk | Contribs) 08:17, 24 February 2007 (UTC)
The problem I see as being this: Just because "a large collection of grains minus one grain makes a heap", that in no way entails that "a large collection of grains minus one grain makes a large collection of grains". And Just because "a large collection of sand makes a heap", that doesn't entail that "a heap is necessarily made by a large collection of sand". There is no logical reason why removing one grain of sand from the heap should necessarily force us to admit that the heap is still a large collection of grains of sand. And if the heap is nolonger a large collection of grains of sand, then if we were to remove one further grain of sand, we wouldn't be logically bound to assert the product to be a heap either. 00:19, 27 February 2007 (UTC)jonbeer
Correct. Your argument laid out as above, where you have split things into "heaps of sand" and "large colections of grains of sand" does not form any kind of Sorites paradox. In fact, it is just a pair of simple statements, with no particularly interesting results. I didn't point this out the first time because I wasn't sure what your point was, but your claim Repeated applications of Premise 2 (each time starting with one less number of grains), eventually forces one to accept the conclusion that a heap may be composed by just one grain of sand is not actually true. The premises can't be linked together in a chain like in the original Sorites paradox, because each of them makes statements about properties of large collections of grains of sand, but neither of them make statements about what is a large collection of grains of sand. You can decide, arbitrarily or by other premises, what constitutes a large collection of grains of sand, but with your premises as they are, that choice is never dictated to you in the way that it is with heaps in the original paradox. Maelin (Talk | Contribs) 04:38, 27 February 2007 (UTC)

This is related, so I place it under here. It seems like something ought to be said about what "makes" means in the article I see possible definitions of make.

  1. a "large collection of sand" is equivalent to a heap. (large collection of sands <=> heap, which means the paradox arises).
  2. "a large collection of sand" can be called a heap (large collection of sands => heap, which do not form the paradox, as premise 2 only talks about "large collections of sands", not heaps.)
  3. A third meaning could be that 'heaps can be called "a large collection of sand"' but I don't see how in any way the sentence can be interpreted that way, so no #3.

I'd add something, but I don't know what philosophers have said about this particular paradox, and I know that in this instance wording DOES matter. Root4(one) 17:23, 12 April 2007 (UTC)

Removed section[edit]

I have removed the section on "visual definition". It didn't contain any citations and isn't part of standard philosophical discussion of this problem - not surprisingly, because it is specific to actual heaps of sand, instead of the abstract concept the paradox is really about. It was added by, and seems to be the original research of, a user whose only other editing activity consisted of adding a similarly questionable section to Monty Hall problem, getting into a flame war over it, and eventually being banned and restored. The content related to the actual paradox seems to be covered adequately in other sections. 15:13, 22 March 2007 (UTC)

I have no problems with this action. I did not introduce the section but, I say again, I thought it was an interesting take on the paradox, and I did work on it to try to make it article worthy. It was an attempt to avoid the paradox by radically defining what something means to be a heap (and I attempted to show that just by radically redefining a heap (in this way) does not allow one to escape the paradox). But in the end, that was MY Original Research as well, and I don't know that I could have gotten the research together or citations to prove that some philosopher somewhere had not similar thoughts and published them.
Root4(one) 16:44, 22 March 2007 (UTC)

Heap is about shape[edit]

I think "heap of sand" is a bad example, because "heap" is just a description of the shape. You could say a heap is a collection of objects, sufficiently numerous that they can't be counted in a single glance, and stacked together with more towards the centre and fewer towards the outside.

The number of items required for a heap is reasonably well defined, it must be about five or six, because you need to be able to stack them in a heap shape, and must not be able to count how many there are in a single glance. People can count up to about five or so items in a single glance and you need at least about three or four items to get the heap shape. You could easily have a heap of clothes or bowling balls with as few as five items. Even if you debate the second criterium I've offered you still can't drop below three and become a heap. No arrangement of two items has the appropriate shape necessary for a heap.

In fact the description of the shape of a heap seems like a better example of the paradox than the number of items in it. How flat does the heap have have to become before it's no longer a heap?

I'm not debating the paradox here, I'm just pointing out that the "heap of sand" example is not a particularly good one and we could probably find a better one. Cheers! 13:12, 25 August 2007 (UTC)

Heap is used because that's the classical example. You'll find the same thing in the Hawk and Dove article: even though the most commonly used explanatory variant of that game is Chicken. Illisium (talk) 07:11, 29 June 2009 (UTC)illisium
Its not just that, although I agree its about shape. But *number is irrelevant* except that below a certain number the shape is impossible. But you can fail to have a heap of sand with a billion grains, if the shape is wrong. Consider an infinite plane of sand one grain thick (being a plane, after all). It is *not a heap*, despite having an *arbitrarily large number of grains*. I think you'd find that other possible phrasings of the paradox also fail because there's a mistaken first premise (that just by having a lot of something it becomes a new class of object), but I can't specifically induct over all possible examples without seeing others. -- (talk) 15:30, 30 October 2012 (UTC)

The article as of late is much clearer[edit]

I just wanted to complement whoever's done the recent edits. A+ on clarity. Root4(one) 16:31, 7 January 2008 (UTC)

Problem solved[edit]

You know, it's not that hard of a problem. Things aren't defined by quantity, but instead by quality. A laptop is defined by its characteristics: two sections: a screen that moves for easier storage when put away, and the control and hardware section, containing the keyboard, mouse, and hardware. Similarly, a heap isn't defined by the number of things making up that heap but the relation of things to each other. A heap is a large pile of small things. Defining large and small is a bigger problem than this. --Guugolpl0x (talk) 19:29, 3 November 2008 (UTC)

This shows some interesting insight into the meaning of the word "heap". But it really has not bearing on the paradox. The paradox presents a number of inconsistent propositions. This "solution" doesn't deny any of them. It doesn't say which one is false and why. So, it's no solution at all.

No paradox - Just imprecise definitions[edit]

It seems obvious that you cannot use something with an imprecise definition to precisely measure or categorise something else. It may be a problem, but it is certainly no paradox. -- (talk) 00:00, 19 June 2009 (UTC)

Well said. There is no paradox because 'heap' is an imprecise definition. This is alluded to in the 'group consensus' section, but saying that the sand has a 'probability of being a heap' is nonsensical. Whoever came up with this paradox obviously had no understanding of how languages evolve and how definitions are learnt. (talk) 12:48, 30 June 2009 (UTC)
Agreed there is no 'paradox', instead it is a worthy example of the influence of semantics on perception. All it really tells us is a that an apparently logical definition can have absurd results (the unmentioned fact that the rules allow you to have a 'negative heap' which most people would call a hole in the beach...) Its greatest value is in how we view things like definitions of infinity (e.g. that which shows the infinity of integers is 'smaller' than the infinity of real numbers). Stub Mandrel (talk) 10:09, 4 July 2016 (UTC)

ZERO grains of sand is a "heap of sand"(?!?)[edit]

"1,000,000 grains of sand is a heap of sand (Premise 1) A heap of sand minus one grain is still a heap. (Premise 2) Repeated applications of Premise 2 (each time starting with one less grain), eventually forces one to accept the conclusion that a heap may be composed of just one grain of sand."

The last sentence stops one short. If 1 grain is still a heap, then according to Premise 2, that "heap (1 grain) of sand minus one grain is still a heap", so 1 grain minus 1 grain = 0 grains which is still a heap, by the above premises.

A "heap of sand" cannot logically be a heap of sand if it contains no sand, so, ipso facto, the wording of the premises must be flawed. (talk) 13:56, 18 September 2009 (UTC)

Original research on accepting the conclusion[edit]

I somehat hastily removed the following comment from the article:

&lt;!-- And, if you follow premise 2 again, composed of a negative number of grains of sand, possibly antimatter. -->

Dromioofephesus pointed out a logical consequence of accepting the conclusion here. Coming from a Wikipedian, it's WP:OR, of course, but this is the first time I've seen this particular argument. I'll be awarding a barnstar for anyone who can source it. Paradoctor (talk) 11:08, 4 March 2010 (UTC)

Found a ref!
Why Induction Is No Cure For Baldness, Yuval Dolev
Article is not accessible online, but Google: sorites paradox "negative number" yields this article as result, with snippet: “person having a negative number of hairs on his scalp.”
…so I think this satisfies sourcing.
—Nils von Barth (nbarth) (talk) 23:12, 18 March 2010 (UTC)
I like it. :) Paradoctor (talk) 23:28, 18 March 2010 (UTC)
Glad to please, and thanks for the star! (…and for requesting this!)
—Nils von Barth (nbarth) (talk) 02:11, 19 March 2010 (UTC)
And as I come to see the butterfly effect of my comment more than four years later, I find that it eventually led to something with a reference. Jolly good, Wikipedians. --Dromioofephesus (talk) 01:59, 18 April 2013 (UTC)

Translating two languages, words and mathematics[edit]

I see the paradox as an attempt to interpret a situation using two languages, one is words the other is mathematics.

The pile of sand is described as a heap. Heap translated into mathematics might be 1,000,000 grains of sand. Repeatedly subtracting one grain a sand away from the heap until there is but one grain left and translating the result from mathematics to words, obviously there is a change in meaning.

Pauljalexander (talk) 06:47, 23 April 2010 (UTC)

Welcome to Wikipedia. The approach of considering the paradox as arising from a translation problem is interesting. Do you happen to know a reliable source we can cite this to? Paradoctor (talk) 16:06, 23 April 2010 (UTC)

I don't have a reference for you but the basis for this idea came from many texts some include wikipedia pages on the [Philosophy_of_language], [Semiotics], [Linguistic_determinism] and [Theory_of_descriptions], writing of Alfred Korzybski and Bertrand Russell. Both of these authors describe mathematics as a language. Heap stands out as not being part of the mathematics language. Is the above argument not clear?

Pauljalexander (talk) 04:20, 28 April 2010 (UTC)

Yes there is sense in your argument, I would further refine it as trying to map from a continuum (number of grains of sand) to a discrete set of states (heap or not heap). Anyway wikipedia need to be well referenced these days, so to include something on this topic we would need to find a good reference so that we dont fall foul of the original research policy.--Salix (talk): 10:14, 28 April 2010 (UTC)

In addition to the contribution above is the following. If half the heap is removed, a half heap would remain, removing half the heap again, would leave a quarter heap. Repeating this until there is but one grain would leave a fraction of the heap. This 'fraction of a heap' is a concept that needs little interpretation and no translation to or from mathematics is needed, because the concept of a heap is tied with the mathematical concept of division and fractions. A reference to quotation that Mathematics is a language,

Pauljalexander (talk) 06:24, 29 April 2010 (UTC)

A heap minus a heap[edit]

It would seem the following would be true

1,000,000 grains of sand is a heap of sand (Premise 1) A heap of sand minus one grain is still a heap. (Premise 2)

One grain of sand is a heap of sand. (Conclusion)

Hence Premise 2 can be rewritten:

A heap of sand minus a heap of sand is still a heap.

I think that clearly shows the fallacy of Premise (talk) 14:46, 9 August 2010 (UTC)

Add a heap of sand to a heap of sand, what do you get? Since addition has an inverse, it follows that the rewritten premise two is actually a theorem of group theory. ;) Paradoctor (talk) 15:57, 9 August 2010 (UTC)


Could someone knowledgeable flesh out the Supervaluationism resolution section? It only describes the "Pegasus likes licorice" statement, and not how it applies to the paradox. I don't understand the connection based on what's written. (talk) 08:40, 11 December 2010 (UTC).

I agree; and I also don't see a connection with Milo's ability to lift a bull (see also section) (talk) 18:49, 17 March 2013 (UTC)
I agree as well. As it stands, the connection between supervaluationism and the sorites paradox is not explicit. If nobody can make it explicit, the section should be removed.

What does "Pegasus fails to refer" mean? Seems like an incomplete sentence. That sort of shorthand is fine, perhaps, for the notational representations, but in the narrative format, it is confusing. What is it that Pegasus fails to refer to? A person? An actual person? What sort of failure? Thanks. (talk) 18:53, 13 November 2013 (UTC)

Is that better? Paradoctor (talk) 21:14, 13 November 2013 (UTC)

Depends on the sand[edit]

Why, clearly a heap of sand is only a heap if the heap-invariant somehow holds. Of course, in that case there must be some way we can order the grains. Ay, there's the rub, for what makes a grain of sand greater or lesser than another? Little which to me makes philosophical sense, but perhaps somebody else would have a better idea. Filburli (talk) 01:50, 26 March 2013 (UTC)

Priest's inclosure schema?[edit]

This is not really my field, but there seems to be quite a bit of current philosophical discussion regarding sorites and Priest's inclosure schema, and whether or not there is a good case for sorites fitting within it cf. (20040302 (talk) 10:11, 28 April 2014 (UTC))

Please stop trying to resolve Sorites within the scope of an encyclopaedia.[edit]

It's not the purpose of an encyclopaedia to resolve Sorites (or any other paradox), but to describe it, it's history, explain why it is important, and possibly refer to different philosophers who have attempted to resolve it, (and their arguments).

But the talk page (and sometime the article) has been filled with attempts to resolve (or defuse, deflate, dissolve) Sorites. (20040302 (talk) 10:19, 28 April 2014 (UTC))

Actually, could an experienced editor not create an archive of 'resolutions', to prevent the article talk page from being flooded by samesuch? (20040302 (talk))
Are you kidding? It's all peace and quiet in here. If you want to see flooding, take a peek at the 37 archive pages of Talk:Monty Hall problem. Face-wink.svg Paradoctor (talk) 10:34, 28 April 2014 (UTC)
OMG yes. Although you haven't objected to my thoughts, but adequately demonstrated that it's a more systemic problem! (20040302 (talk) 10:42, 28 April 2014 (UTC))
Asking whether someone is kidding is generally considered to imply an objection to what has been said. 19 short sections in 10 years means this is a very quiet place, absolute as well as in relation to page views. Really. Face-wink.svg Paradoctor (talk) 11:40, 28 April 2014 (UTC)


1: When does the collection of tiny pebbles become a heap of stone?
[When does the heap of stone become a few pebbles, or even a single pebble?]
2: ...
3: The difference between the consecutive "large" heaps here is twice the different between consecutive "small" heaps, but is less significant.
4: The difference between the consecutive "large" heaps here is twice the different between consecutive "small" heaps, but is less significant.
5: The change in area between consecutive "big" heaps (left) is more than thirty times that of the change between consecutive "little" heaps (right), yet seems less significant.

User:Jochen Burghardt left me the following message regarding the [first] image:

In the article Sorites paradox, I consider the basic idea of your picture File:Sorites paradox heap.png useful, but I agree with McGeddon that "1=1+1=..." shouldn't appear there. I see two issues:
  1. use of "=" suggests that all heaps are claimed to be the same, while the paradox is only about all have a common property, viz. being a heap;
  2. the psychological temptation to claim "one pebble more or less can't affect the heap property" works better for sufficiently large heaps than for small nonheaps (collections of 2 and 9 pebbles "look quite different", while such of 100002 and 100009 don't).
Maybe it is possible to design a picture starting with the large heap, then somehow indicating the removal of the first 1, 2, 3 pebbles. To avoid "=", e.g. a heap/nonheap border could be indicated somehow. However, I don't know how to illustrate that it's a problem which collections are immediately left and right to the border - maybe you have an idea.

Hyacinth (talk) 22:38, 8 October 2014 (UTC)

I numbered the images to ease referring to them. In image 2, I still have the problem with "=". I think image 3 and its caption illustrates an important aspect of the paradox; but I'm not sure whether it is the main aspect (because I'm not really sure what the paradox' main aspect is...). I preferred your first version, without parantheses; in the actual version they are much bigger than the small heaps, which can be confusing. You could use instead e.g. red color for the dots ("...") to indicate that they are not heaps, but kind of meta symbols here. And the small heaps are hardly visible, perhaps you could decrease the image width by showing only 4 large and 4 small heaps, i.e. omitting the smallest 6 heaps. And you could clip almost all left and right white border. - Jochen Burghardt (talk) 13:31, 9 October 2014 (UTC)

Picture 4 shows what I had in mind. However, it doesn't convince me either: contrary to what the caption claims, the small heaps still don't look more different in size than the large ones. - Jochen Burghardt (talk) 06:57, 10 October 2014 (UTC)

New image, picture 5. Hyacinth (talk) 23:41, 5 March 2015 (UTC)

looking at apparent change between heaps of different volume appears, to me, to be a discussion of perception based on the [Surface-area-to-volume ratio] of a heap, rather than anything to do with sorites, which is to do with identifying the presence oo absence of 'heap hood' - though I admit, I may be missing something here 20040302 (talk) 09:34, 6 March 2015 (UTC)
The problem with the captions to images labelled 3 and 4 on the talk page is that it's not at all clear what measure is being used. "Difference between the heaps" -- in what? Size? Number of particles (which difference is, by hypothesis, one grain and therefore the same for each pair of heaps n, n+1). For #5 you might be trying to convey that a difference of one unit in the length/width of the square gives rise to an (inverse) square change in the area, but I'm still not convinced the math ("thirty times") works here or even if it does, if it's germane to the Sorities argument. (talk) 17:20, 18 March 2016 (UTC)
Agreed, none of these images seem to illustrate the paradox at all. Image three is closest as an illustration of "the heap stops being a heap somewhere along this line - but where?", but it doesn't really seem any clearer than one photo of a heap of sand. I'll shovel something up from commons. --McGeddon (talk) 18:19, 18 September 2016 (UTC)
Color doesn't have much to do with "heaphood". Hyacinth (talk) 22:27, 29 March 2017 (UTC)

Great Article[edit]

I just want to say this article is fantastic. I learned a lot from it and it's very interesting, well written too. I would advise against making major changes with it. EggsInMyPockets (talk) 17:55, 29 March 2017 (UTC)

etymology missing[edit]

i missed the meaning of the word from the article. an etymology would be useful to add, like here you can find:

or at least the word sorites needs to turned int a link - there is a relevant article about what sorites is. (talk) 14:41, 27 May 2017 (UTC).

okay i put in the link though its not the first instance of the word, instead the text under the picture. (talk) 14:50, 27 May 2017 (UTC)

I removed your link, since it pointed to the disambiguation page Sorites, where the only sensible link (in our context) was back to Sorites paradox. I agree that it is a good idea to state somewhere here what 'Sorites' literally means, and when it was coined. - Jochen Burghardt (talk) 16:37, 27 May 2017 (UTC)
This text is already present, under Sorites_paradox#Paradox_of_the_heap. If you have additional useful etymological information, best place it there. - Jochen Burghardt (talk) 16:42, 27 May 2017 (UTC)

suggested resolution[edit]

I know there have been various suggested resolutions before, but can't this be resolved by pointing out that a heap of sand is not actually a measure of quantity, and is basically another word for a pile of sand? or, indeed, mound of sand? it's not exactly something that needs a precise definition. (That, and it occurs to me that this paradox comes under the same category of paradox as Zeno's Paradox, and the one surrounding imprecision in coastline lengths. Namely that it is pretty much exclusively a purely philosophical paradox- if you try to argue the point in a non-philosophical setting, you will just be told to shut up.)Sstabeler (talk) 14:17, 13 September 2017 (UTC)

I think it was W. V. O. Quine who pointed out that the Sorites patadox applies to almost all our notions, comprising adjectives (e.g. red, large, cold, stormy) as well as nouns (e.g. river, chair, dog), and, that we often understand a sentence despite the vagueness of all its components by taking them to be meant relative, not absolute (e.g. the red chair refers to the object that satisfies both red and chair more than the remaining objects in the scene). (If I'd find the reference, I'd like to add this to the article.) Whoever thinks e.g. about computer software for natural language understanding has to consider such issues. - Jochen Burghardt (talk) 16:25, 13 September 2017 (UTC)

Merger proposal[edit]

@Nikola Smolenski, Gregbard, Nbarth, Mesoderm, Charlotte Aryanne, Jeraphine Gryphon, Sandra lafave, and Raven530:

I suggest to "merge" the article "Continuum fallacy" (CF) into here. Actually, I believe this would amount almost to a deletion of "Continuum fallacy", as it doesn't provide much information that isn't already contained in Sorites paradox (SP). I suggest to keep the following parts.

The fallacy is the argument that two states or conditions cannot be considered distinct (or do not exist at all) because between them there exists a continuum of states.

Narrowly speaking, the sorites paradox refers to situations where there are many discrete states (classically between 1 and 1,000,000 grains of sand, hence 1,000,000 possible states), while the continuum fallacy refers to situations where there is (or appears to be) a continuum of states, such as temperature – is a room hot or cold? Whether any continua exist in the physical world is the classic question of atomism, and while Newtonian physics models the world as continuous, in modern quantum physics, notions of continuous length break down at the Planck length, and thus what appear to be continua may, at base, simply be very many discrete states.

For the purpose of the continuum fallacy, one assumes that there is in fact a continuum

They could be added to Sorites paradox#Variations where the continuum fallacy is already mentioned; some copyediting may be necessary.

None of the remaining (CF) examples (beard, sand) and references (Roberts, LaFave) deals with a continuum in the mathematical sense; the same applies to the Thouless reference already present here (SP). For this reason, I suggest to add a warning that in popular philosophy "continuum" is often abused for "too large to glance over". This would nicely fit as a footnote after "(or appears to be)" in the 2nd part. - Jochen Burghardt (talk) 09:17, 18 April 2019 (UTC)