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Under the assumption that removing a single grain does not turn a heap into a non-heap, the paradox is to consider what happens when the process is repeated enough times: is a single remaining grain still a heap? If not, when did it change from a heap to a non-heap? (Wikipedia, "Sorites paradox")

Since paradoxes don't actually exist, the sorites paradox is predicated on a faulty assumption. A heap is defined as a set of objects where some of the objects in the set prevent other objects in the set from reaching a lower potential energy. This means removing a single object can turn a heap into a non-heap even if there are still objects remaining in the set. Those remaining objects would be the bottom layer of the now-not-heap.

Frank Hubeny
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Hierarchist
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    "Since paradoxes don't actually exist" ??? Paradoxes exist. – Mauro ALLEGRANZA Apr 08 '19 at 14:22
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    The difficulty is exactly to define "heap" : it is not defined quantitatively : so many grains... Thus, the paradox arises when we apply a typical quantitative apprach: specifically a sort of "Mathematical induction": an heap minus 1 grain is still an heap. Thus, by induction, whatever is the number of grains removed, the result (also when empty) is still an heap. Contradiction. – Mauro ALLEGRANZA Apr 08 '19 at 14:25
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    Sorites paradox is related to the issue of so-called Vague predicates. – Mauro ALLEGRANZA Apr 08 '19 at 14:26
  • @MauroALLEGRANZA, "heap" is defined as a set of objects where at least one object is prevented from reaching a lower potential by the other objects in the set. Paradoxes don't exist. 3 apples don't exist when I add one apple to another apple just because I don't define two one apples as not three apples. Regardless, you did answer my question: "yes, it's a paradox only because someone didn't know what a heap is." – Hierarchist Apr 08 '19 at 14:53
  • Not clear what a "heap" is for you... heap : "an untidy collection of objects placed haphazardly on top of each other." But the same for a pile of cards one exactly on top of the other: we start with a pile of 1.000 cards and remove the first one (the one on top) : obviously, what is left is still a ple. And so on... – Mauro ALLEGRANZA Apr 08 '19 at 15:01
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    The sorites paradox doesn't just apply to heaps. It also applies to predicates such as 'tall', 'bald', 'big', 'old', 'red', and so on. The paradox shows that natural language predicates are not precise. Trying to give precise definitions to these words is hopeless. – E... Apr 08 '19 at 15:31
  • @MauroALLEGRANZA, "on top of each other." Exactly. – Hierarchist Apr 08 '19 at 15:36
  • @Eliran, you are "tall" to someone if someone believes you are "tall". If take an inch away from your height, will that someone still believe you are "tall"? This is called informational qualia. – Hierarchist Apr 08 '19 at 15:38
  • Ok, problem solved. We should let all the philosophers know. – E... Apr 08 '19 at 15:40
  • @Eliran, no problem was solved, because there was never a problem. To solve a problem, it has to exist first. But it appears that we've cleared away your confusion. – Hierarchist Apr 08 '19 at 15:41
  • "It also applies to predicates such as 'tall', 'bald', 'big', 'old', 'red',", 'believe', ... – H Walters Apr 08 '19 at 15:59
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    "Since paradoxes don't actually exist", how do you figure that? Of course they do, it happens when different trusted principles clash. In this case, modus ponens and induction applied to vague terms like the "heap". But it works the same with colors, or any other accumulation of small changes, so potential energy and other specifics are irrelevant. When it comes to vague terms one just has to give up modus ponens or induction (commonly, the latter). – Conifold Apr 08 '19 at 21:14
  • Yes! If you properly define "heap", then you could say exactly when it stops being a heap. Does that make the paradox go away? NO! Because the problem is not with the heap, it is with the vagueness of the term before you REDEFINED it. The vagueness didn't go away: it still exists in other weakly defined concepts; as long as words like 'tall', 'bald', 'big', etc. are not precisely defined for every instance they are used, the sorites will stay. Can we have language without imprecise terms? Unknown - we will have to redefine, one by one, until the the vagueness stops being vague... – christo183 Apr 09 '19 at 06:47
  • @christo183, the arbiter of the vague term is the subject. This means solely the subject decides when a short tree becomes a tall tree. And because perception of the growing short tree does not happen every instance, the subject does not need to define "tall tree" to the nanometer. The subject must merely look away for a week or two. Paradoxes don't exist. – Hierarchist Apr 09 '19 at 15:04
  • @Hierarchist Now you are contradicting yourself: The subject would need a strict definition of "tall" in order to be able to perceive when the short tree became tall - this is the case where there is no paradox. (Okay "every instance" is a bit strong, every application is more accurate). But all that is beside the point: Can we have language without vagueness? If your answer is "yes", I would like to see your proof. If you answer "no", well others have already explained why that is the problem. – christo183 Apr 10 '19 at 06:45
  • The subject does. Either the subject thinks "X is tall" or the subject does not. Where is the contradiction? You don't even need non-vagueness, you just need the vagueness to be below our ability to perceive change. You don't watch a tree grow 24/7 for a decade and then get a moment of epiphany "the tree is tall now". You keep the tree out of your perception for long enough to jump over that unperceived line. The line between "tall" and "not tall" is unperceived because we do not practice perception 24/7. – Hierarchist Apr 11 '19 at 02:00
  • I always wonder why its called the Sorites paradox. It's not a paradox but just a definitional problem. 'Heap' is a vague term, as is 'tall', bald' and so forth. A vague term is not paradoxical. I'd want to ask on what grounds the Sorites problem is called a paradox. –  Apr 11 '19 at 10:57
  • @PeterJ because removing a single nanometer does not cause something tall to become short, repeating this process over and over will never cause said something tall to ever become short. It is a logical confusion since "X is tall" is not BASED on how many nanometers tall X is, but on what the perceiver (the originator of "X is tall") thinks. Our thinking is not based on nanometers, thus undermining the whole logical confusion. – Hierarchist Apr 11 '19 at 13:09
  • @Hierarchist - Yes, this is why the problem goes away as soon as we define 'heap' as a certain minimum quantity. There is no paradox, just a loosely-defined term. –  Apr 11 '19 at 15:29

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The sorites paradox as Wikipedia describes it offers a clue:

The sorites paradox (sometimes known as the paradox of the heap) is a paradox that arises from vague predicates.

The clue is "vague predicates":

In philosophy, vagueness refers to an important problem in semantics, metaphysics and philosophical logic. Definitions of this problem vary. A predicate is sometimes said to be vague if the bound of its extension is indeterminate, or appears to be so. The predicate "is tall" is vague because there seems to be no particular height at which someone becomes tall. Alternately, a predicate is sometimes said to be vague if there are borderline cases of its application, such that in these cases competent speakers of the language may faultlessly disagree over whether the predicate applies. The disagreement over whether a hotdog is a sandwich suggests that “sandwich” is vague.

If we accept Michael Polanyi's view that

...all knowledge is either tacit or rooted in tacit knowledge (page 7)

one should not expect to obtain a complete explicit knowledge description of heap.

Here is the question:

Isn't the sorites paradox predicated on a non-understanding of what a heap is?

If one views "non-understanding" as not having a complete explicit knowledge of what a heap is, but only tacit knowledge, then this would characterize the problem.

Polanyi, M. (1966). The logic of tacit inference. Philosophy, 41(155), 1-18.

Wikipedia contributors. (2019, March 6). Sorites paradox. In Wikipedia, The Free Encyclopedia. Retrieved 17:32, April 8, 2019, from https://en.wikipedia.org/w/index.php?title=Sorites_paradox&oldid=886399081

Wikipedia contributors. (2019, March 4). Vagueness. In Wikipedia, The Free Encyclopedia. Retrieved 17:34, April 8, 2019, from https://en.wikipedia.org/w/index.php?title=Vagueness&oldid=886065402

Frank Hubeny
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    One, a heap is explicitly defined so you're right the only problem is that someone didn't know what a heap is. Two, only having tacit knowledge, like the heap is determined to be a big heap, does not create a paradox either, since the shrinking heap is a "big heap" for as long as the subject perceives it as such since the very concept of a "big heap" was created by the subject in the first place. It stopped being a big heap when the subject proclaimed it so. To say there's a paradox is to try and turn tacit knowledge into explicit knowledge. – Hierarchist Apr 08 '19 at 19:05
  • @Hierarchist Your definition of heap doesn't define heaps very well. It certainly applies to stacks, and arguably applies to anything that has vertical extension. – Willie Betmore Mar 26 '20 at 00:31
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It may help to think of sorites as less of a singular, concrete paradox, and more of a strategy for constructing paradoxes. For example, the Stanford Encyclopedia article about Transworld Identity describes this seemingly unrelated paradox in the field of modal logic:

One such argument, adapted from Chisholm 1967, goes as follows. Taking Adam and Noah in the actual world as our examples (and pretending, for the sake of the example, that the biblical characters are real people), then, on the plausible assumption that not all of their properties are essential to them, it seems that there is a possible world in which Adam is a little more like the actual Noah than he actually was, and Noah a little more like the actual Adam than he actually was. But if there is such a world, then it seems that there should be a further world in which Adam is yet more like the actual Noah, and Noah yet more like the actual Adam. Proceeding in this way, it looks as if we may arrive ultimately at a possible world that is exactly like the actual world, except that Adam and Noah have ‘switched roles’ (plus any further differences that follow logically from this, such as the fact that in the ‘role-switching’ world Eve is the consort of a man who plays the Adam role, but is in fact Noah). But if this can happen with Adam and Noah, then it seems that it could happen with any two actual individuals. For example, it looks as if there will be a possible world that is a duplicate of the actual world except for the fact that in this world you play the role that Queen Victoria plays in the actual world, and she plays the role that you play in the actual world (cf. Chisholm 1967, p. 83 in 1979). But this may seem intolerable. Is it really the case that Queen Victoria could have had all your actual properties (except for identity with you) while you had all of hers (except for identity with her)?

[...]

Chisholm (1967) arrives at his role-switching world by a series of steps. Thus his argument appears to rely on the combination of the transitivity of identity (across possible worlds) with the assumption that a succession of small changes can add up to a big change. And ‘Chisholm’s Paradox’ (as it is called) is sometimes regarded as relying crucially on these assumptions, suggesting that it has the form of a sorites paradox (the type of paradox that generates, from apparently impeccable assumptions, such absurd conclusions as that a man with a million hairs on his head is bald). (See, for example, Forbes 1985, Ch. 7.)

[goes on to describe a variation of the argument which is not sorites-like, by doing the whole thing in a single step]

Italics in original, boldface added for emphasis.

The broader context of this paradox is the question of whether we should identify Noah-in-world-1 with Noah-in-world-2 (i.e. whether they should be considered "the same" Noah or "different" Noahs). This paradox argues that they are different, by suggesting that we may construct the scenario such that worlds 1 and 2 are identical except that Noah and Adam have "switched places." Then Noah-in-world-1 and Adam-in-world-2 are functionally the same in every particular, despite officially being "different people." More broadly, worlds 1 and 2 are functionally the same in every particular, with only the identities of Adam and Noah differing. This is unsatisfactory for a variety of reasons. The only alternative resolution is to claim that Noah (or Adam) has some "essential Noah-ness" (resp. "Adam-ness") that does not change across possible worlds. That is unsatisfactory (or at least questionable) for an entirely different set of reasons. See the linked article for a fully contextualized discussion of these issues.

In this context, the sorites form of the paradox (in which we construct a series of worlds where Noah is progressively more like Adam and vice-versa) serves to emphasize the fuzzy edges of the standard possible world semantics. While the sorites form may be a little weaker, in that it requires additional assumptions and a more complicated construction, it is more "obviously correct," and provides a deeper intuition into why the non-sorites version is also correct. That's why Stanford leads with the sorites form of the paradox before simplifying it into the non-sorites form.

Another famous variation of the sorites paradox is the Ship of Theseus, in which a ship's planks are replaced one at a time, but it's still "the same ship." Curiously, many of these arguments seem to revolve around identity, probably because we like to think of identity as a binary, fixed relation. We are generally uncomfortable saying that A is 85% identical to B; either they are identical or they are not. This discomfort provides a fertile ground for logical inconsistency of many different forms. The sorities technique, then, can be used to build those inconsistencies up into paradoxes, which helps to define and shape the theories we construct in order to cure the underlying inconsistencies.

In this case, I fear there may be an additional misunderstanding at work. OP has, in both the question and in several comments, asserted that:

Paradoxes don't exist.

I think what the questioner means by this assertion is the following:

Flaws in reality that create contradictions don't exist.

This is, so far as we're aware, perfectly true. But it's also irrelevant, because philosophers normally define a "paradox" as a flaw in our understanding of reality, rather than a flaw in reality itself. So, for example, the sorites paradox does not mean that there is some problem with actual heaps of sand that causes them to behave oddly when we remove one grain at a time. Rather, it means that there is a problem with our definition of the word "heap." The particular definition which the questioner supplies might be one possible resolution to the paradox, but it does not obviate the paradox itself, because there could be other definitions of "heap" which might resolve the paradox in a different way.

(It would, of course, be quite absurd to assert that there are no flaws in our understanding of reality, so I have attempted to guess the questioner's true meaning. I might have guessed wrongly.)

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Kevin
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  • The Ship Of Theseus is not a paradox (paradoxes don't exist). From its construction, the concept "the ship of Theseus" is created by some subject, or group of subjects. This means the validity of "X is the ship of Theseus" is wholly dependent on the decision of those same subjects. Trying to say there is no way to know if X is/isn't the ship of Theseus would be to forget where the very concept of "the ship of Theseus" came from. An appeal to willful ignorance is a Logical Fallacy. Analogy: "my salad" is in a bowl and has tomatoes in it. If I eat all the tomatoes first, it's still "my salad". – Hierarchist Apr 10 '19 at 23:44
  • @Hierarchist: I have attempted to guess what you mean by "paradoxes don't exist" and edited my answer accordingly. – Kevin Apr 11 '19 at 00:26
  • I would endorse this answer. There is no paradox, just a some vague definitions. There's nothing to prevent us defining 'heap' as a minimum of 250 grains and then the problem goes away. I like your point that it is not a paradox but a strategy for creating them. . –  Apr 11 '19 at 11:04
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You're right that it's based on a faulty assumption. Invoking vagueness is not the only way out of this. Your definition can probably be refuted somehow, but no matter what the definition, mathematics assures us that at some point you have a heap such that the next grain of sand you remove makes it fail to be a heap, provided you assume that even if you can't determine whether or not it is a heap yourself, a collection of sand is either a heap or not a heap.

It's actually quite puzzling that people would accept the premise that removing a single grain of sand can't turn a heap into a non-heap, and the Sorites "paradox" should really just be seen as a proof that the premise is false.

Matt Samuel
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  • "You assume that even if you can't determine whether or not it is a heap yourself, a collection of sand is either a heap or not a heap" is the position of epistemicists. It is a bit comical, however, and is held by very few. There is no "can't" with determining heaps, since the notion is our own creation, there is simply nothing there to determine. Careful epistemicists admit that "not being able to determine" is a pretense. But taken seriously, epistemicism confuses not being able to determine something with determining not making sense at all. In other words, the above assumption is false. – Conifold Apr 09 '19 at 20:17
  • @conifold But the majority view that it has no truth value explicitly says that there is a point where you can't determine whether it is a heap, and if that's the case then there is also a single grain of sand you remove to bring it from a heap to that state, regardless of how counterintuitive you find it. – Matt Samuel Apr 10 '19 at 16:49
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    This is a standard observation that vagueness can not be removed by introducing additional boundaries, as in between heaps and not quite heaps, because the same problem re-emerges. But epistemicism fails not because of "counterintuitiveness", but because it is clear that there is no source for the supposed "determination" (and pretend epistemicism does not even suggest there is, and rightly so). The problem is with the idea of a fixed meaning for "heap", it just does not have any meaning, and hence truth values, in absolute terms. What is a heap in one context is just a handful in another. – Conifold Apr 10 '19 at 19:34
  • @Conifold Saying a single grain of sand doesn't matter is like saying one vote doesn't matter in a large election. Usually it doesn't make a difference, but it's simply wrong to say that it never does. – Matt Samuel Apr 10 '19 at 19:49
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    No, it is not like. Elections have absolute cutoffs written into the rules, so majorities are not like heaps at all. In the case of heaps, with their context dependent meaning, where the context is addition/subtraction of a single grain, it does never matter, by the sense of the word. The mistake is to assume that the context, and hence the meaning, stays the same all the way through induction. I am not saying that epistemicism can not be a technical workaround when one does wish to fix some absolute meaning without committing to specifics, but it is no explanation of the phenomenon. – Conifold Apr 10 '19 at 20:13
  • @Conifold That's not a solution to the problem, it's an attempt to get around the mathematical fact that for something finite there simply are boundaries, and you cross those boundaries one element at a time. It has nothing to do with the specific definition of the term. If the term has any meaning, the boundary exists. Otherwise you can't say that anything is a heap, which is clearly not true. – Matt Samuel Apr 10 '19 at 20:43
  • @Conifold By the way, the boundary does not depend only on the number. It could be that if you remove this grain of sand it ceases to be a heap, but if you remove a different one it is still a heap, and there's no reason for it to be the same number for different heaps. – Matt Samuel Apr 10 '19 at 20:48
  • There are no mathematical facts about non-mathematical concepts, and the meaning is not restricted to mathematical concepts. It is determined by how the term is used, which does not have to be coherent across different contexts. That is (obviously) the explanation of what happens with "heaps". What you are trying to do instead is replace the colloquial meaning with something mathematical, and, no surprise, it simply does not fit very well. Such replacing can be useful for some technical purposes, but has nothing to do with explaining the paradox itself. – Conifold Apr 10 '19 at 21:17
  • @conifold If something can't be explained by mathematics, then it is a heap of nonsense, or at least completely devoid of reason. The paradox itself uses axioms and logical deduction and therefore is mathematics. You can't then say "Never mind, mathematics doesn't apply because I said so." – Matt Samuel Apr 13 '19 at 19:58
  • The first sentence is obviously false since most of science is not mathematized. Axioms and deduction are used to flesh out what the paradox is, and in particular, to show that if one wishes to define a "mathematized" version of heaps one would have to give up their standard meaning. – Conifold Apr 14 '19 at 00:13
  • @MattSamuel Who was the first person to speak English? Mathematically, there must have been such a person, right? How did the other people who spoke Anglo-Saxon understand what she was saying? – Willie Betmore Mar 26 '20 at 00:55