Talk:Sorites paradox

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[edit] Ummmm....

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  • Premise 1 and 2 are no longer applicable, unless we can conclude that a heap is somehow related to a large collection of grains of sand.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.

[edit] Suggested Definition

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)

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. http://en.wikipedia.org/w/index.php?title=Talk:Sorites_paradox&action=submit#

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)

[edit] A case for name change?

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. 70.111.248.60 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)

[edit] Case for a merger?

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


[edit] Removed section

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

[edit] Differences in Ages

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 2.1.204.95.23.122 20:50, 24 November 2006 (UTC)

[edit] Three Valued Logic

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)

[edit] There is no paradox

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)

[edit] Minor Error

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.


[edit] Resolutions dubious

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. 80.195.231.43 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? 80.195.231.43 11:38, 7 February 2007 (UTC)


[edit] Flawed Example???

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"?

~dongray2

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. 195.137.109.140 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)

[edit] Removed section

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. 129.97.79.144 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)

[edit] Heap is about shape

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!

80.192.29.107 13:12, 25 August 2007 (UTC)

[edit] The article as of late is much clearer

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