Bear with me here, I know nothing about philosophy that I haven't read on Wikipedia.

I don't understand why the Sorites paradox is considered an unsolved problem in philosophy (according to Wikipedia). I have a proposed mathematical solution.

The premise is that if you remove a grain of sand from a pile of sand, then it remains a pile, and the paradox is that when you remove all of them there is no pile.

Let's say you have a decision procedure that takes as input a collection of grains of sand configured somehow in space and outputs true or false based on whether or not the sand constitutes a pile. It could fail to be a pile, for example, if any two grains are a million miles apart, even if there are many grains of sand.

Each collection of sand that it deems to be a pile has some number of grains. The set of all these numbers is a subset of the positive integers. Therefore it has a least element. Therefore, no matter how you define a "pile" of sand, there is a well defined positive minimum number of grains a collection of sand has to have in order to constitute a pile. Therefore we have refuted the original premise that removing a grain of sand from a pile still leaves behind a pile, no matter how "pile" is defined.

Why would this not resolve the Sorites paradox?

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    It seems like you're merely transposing all of the work the paradox proposes into your decision procedure that is able to deem whether or not something is a pile. A lot is hidden here as well: Therefore it has a least element. Therefore, no matter how you define a "pile" of sand, there is a well defined positive minimum number of grains a collection of sand has to have in order to constitute a pile. – virmaior Aug 18 '16 at 4:42
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    "I have a proposed mathematical solution." A mathematical solution would involve a formal proof that the proposed decision function can be constructed in principle. Without a rigorous proof, there is no reason to think that for any decision function you propose that the smallest collection of grains that satisifies the "is heap decision function" is greater than 1. – Dave Aug 18 '16 at 20:32
  • @Dave Trust me, I know about rigorous proofs (I have a PhD in math) and my argument qualifies. Do you see any corners being cut or anything that cannot be made precise? – Matt Samuel Aug 18 '16 at 22:30
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    If we just remove every grain we should end up with the empty pile..? ;-) – Ben Aug 19 '16 at 6:09
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    @Ben that doesn't address the issue though because the single grain of sand probably isn't a pile. We need to know that there is a pile we can remove a grain from to leave behind a non-pile. – Matt Samuel Aug 19 '16 at 9:07

The previous answers betray a lack of familiarity with the literature. Your solution, using the least number principle, essentially works. It is a known argument for epistemism about vagueness, the position that vague properties have sharp unknowable boundaries. If I recall, it is discussed at the beginning of the last chapter of Timothy Williamson's Vagueness, so it is a fairly central part of the locus classicus for epistemicism. (Edit: beginning of second to last chapter)

But it goes without saying that someone presented just with your solution wouldn't possess a full story about the challenge posed by vagueness. Many logics defeat the Sorites (the main categories are fuzzy, supervaluationist, epistemicist, and contextualist). If you are not an epistemicist about vagueness, you deny the least number principle for the extension of a vague predicate. If this sounds incoherent to you, you aren't alone. You're an epistemicist. However, to understand why very many smart people reject epistemicism, you should look at the literature on supervaluationism (Fine 1975), fuzzy logic, which has proliferated into its own field of model theory, and contextualism (Shapiro 2006 might be the most robust vagueness logic currently on the market). One common position among non-epistemicist logicians who study vagueness is that vagueness refers to various non-Boolean properties exhibited by natural language, like gradability and typicality effects. In my opinion, the epistemicist's best response to the presence of these fuzzy properties is that they are without fail modeled in a classical metalanguage. However, the question whether metalogic should arbitrate disputes in the object language is both murky and high stakes.

For the full vindication of your intuition, see Williamson 1994.


Your proposed solution does not solve the paradox.

The whole point of the paradox is that the term 'pile' is vague. That is, given an object (e.g. a collection of grains of sand) it is indeterminate whether the term applies to this object or not. It is indeterminate since it's not clear just how many grains constitute a heap (for any number n, you can appropriately ask, why not n+1?). And just settling on a number n would be completely arbitrary. Your proposed solution assumes that there is a procedure that decides this.

Natural language is full of vague terms (e.g. tall, bald, young, nice) and the paradox can be replicated with any of them. Some of these (e.g. 'nice') are even more complex in their vagueness, and it is not clear what a decision procedure would look like for them.

For a more in-depth presentation of the paradox and some proposed solutons, you can read the SEP entry on the Sorites Paradox.

  • The decision procedure does not depend only on the number. It can depend on anything you want. Regardless it still has a minimum. There may be configurations of sand that fail to be a pile even if they are above the minimum, but nothing below the minimum is a pile. – Matt Samuel Aug 18 '16 at 13:34
  • In particular, the minimum is not arbitrary. It only appears that way because of the large number of grains involved. As difficult as it is to conceive, there is a dividing line, possibly different for every particular pile. – Matt Samuel Aug 18 '16 at 13:54
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    @MattSamuel What you're implying is that the term 'heap' is determinate, since you can provide a procedure that decides every case. The paradox starts with the assumption that there are borderline cases for the given term. Even if this is resolved for 'heap' it's unclear what a decision procedure for terms like 'nice' would look like. – Eliran Aug 18 '16 at 14:06

This is all about the difference between natural language and formal language. In formal language, a term cannot be used unless it's well-defined according to the standards of the language. In natural language, on the other hand, well-defined terms are the exception rather than the rule.

The Sorites paradox forces us to us to recognize that a term like "pile," which we use commonly and usefully, may seem to be well-defined but is not --in other words that there is no single, universal decision procedure attached to the term "pile" as commonly defined. Your solution proposes to substitute the natural language term with a new, well-defined term (a term with a decision procedure). Although this solves the problem by fiat, it has no bearing on the original dilemma.

However, this is not entirely a bad thing, since modern logic was created by substituting fuzzy natural language concepts with new, well-defined ones.

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    I realized finally this is the right answer. I have a PhD in math. The possibility that "pile" would be impossible to define didn't even occur to me because clearly it's a term we use, and in math we can define everything we use. I realize I've been so immersed in mathematics it has shaped my perception of language and reality in general. There are sentences with "obviously" straightforward, well defined meanings in mathematics that, when shown to a non-mathematician, seem convoluted and ambiguous. For me, these sentences truly have no ambiguity. The... – Matt Samuel Aug 20 '16 at 23:55
  • ...alternative interpretations seem linguistically incorrect. So, yeah. – Matt Samuel Aug 20 '16 at 23:57
  • @MattSamuel: Related to your induction proof and the limits of natural language, you may be interested in analyzing Berry's paradox and the surprise test paradox, both of which are even harder for mathematically untrained people to correctly resolve. – user21820 Aug 22 '16 at 6:32
  • @MattSamuel This is basically what most other answers were getting at. Chris probably managed to present it in the clearest way. – Eliran Aug 23 '16 at 12:44
  • @Eliran Some of the other answers missed my point. They seemed to think that I was saying there is a fixed number, that 45678 grains of sand was always a pile and 45677 is not. That may be true for some pile with 45678 grains, but it could just as easily happen that even though the minimum is 45678 grains there are probably collections with 45678 or more grains that are not piles. And the minimum is not arbitrary, though it'd certainly be hard to figure out. But if pile can't be defined then the argument doesn't apply. – Matt Samuel Aug 23 '16 at 13:16

I agree with some of the other answers that you're going to have a very difficult time spelling out exactly what the decision procedure involved would be. If the decision procedure boils down to you just stipulating that x is a "pile" iff x is composed of 47526 grains of sand (say), then you've haven't solved the problem but only transferred it to a new question.

To see why, consider how an objection is going to go: Presumably being a pile on your account is an objective feature of the world, which some things have and other things lack. So, what I want you to tell me now is what grounds that feature? What is it about grain 47526 that makes the difference which suddenly confers pilehood upon the group of 47525 grains that were there beforehand?

You could try to reply to that objection that it's just a primitive, unexplainable fact that 47526 is the precise number of grains that constitute a pile, but that looks farfetched. How did you come to know that precise number? What test did you perform for "pilehood" that let you determine that was the number?

I think a better strategy for solving the paradox which might be in the same spirit as your proposal above is epistemicism. The view's foremost defender is Timothy Williamson. The basic idea here is that there can be no vagueness in reality; therefore there is a fact of the matter as to whether this pile of grains of sand constitute a heap or not. Rather, Williamson says, vagueness is just epistemic. The pile either is a heap or it isn't; we just can't tell which. (This is an improvement over your suggestion because it doesn't commit Williamson to having to identify a decision procedure that can tell what make the piles piles).

Epistemicism solves the Sorites paradox if it is true. But, of course, whether it is true is a matter of further debate.

  • I never said I knew what the "correct" decision procedure is, or whether I have one at all. It's true for any decision procedure, no matter how convoluted. It could say that 5 grains of sand make a pile if it's Tuesday. Certainly no worthwhile decision procedure would depend only on the number. – Matt Samuel Aug 18 '16 at 20:01
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    But there's going to have to exist some such finite, mechanically followable procedure. And note that just making the procedure more complex doesn't really help with the fundamental problem of grounding the difference between piles and non-piles. Why should having five grains on a tuesday be more explanatory than just plain being five grains? – shane Aug 18 '16 at 20:08
  • I do not understand Williamson's reasoning. "Heap" isn't a natural kind, so even if there is no "vagueness in reality" it does not follow that there are any facts of the matter concerning heaps. The term is used based on loose agreement among language speakers, hence the vagueness. Does Williamson really suggest that there is a threshold for heaps but we don't know what it is, or is it supposed to be therapeutic a la Wittgenstein: don't worry, it's not reality it's just us, bewitched by language? – Conifold Aug 19 '16 at 3:12
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    The decision procedure being feasible is not actually an important part of my argument. In Williamson's philosophy my conclusion is still correct, even if we can't possibly identify when it happens. – Matt Samuel Aug 19 '16 at 12:30
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    @MattSamuel "feasibility" isn't the important part. The important part is "specifiability". Since a decision procedure is an algorithm, and algorithms require a finite number of specified steps, you haven't given a decision procedure until you've specified the rules that the procedure is going to follow to determine heaps from wholes. My objection is that any way you try to provide such a specification is going to just turn out to be stipulative under analysis. – shane Aug 22 '16 at 12:38

Yes, you could "resolve" the paradox that way. But you will have resorted to "Humpty-Dumptyism" in order to do so. Arguably that is even worse than the original "paradox".


Language is necessarily vague. If you remove all vagueness, you are arguably not using language any more. Which is I suppose another way to frame the same paradox.


In natural language vagueness is a useful concept, and we might think that this is all that the paradox is pointing to.

However, I take the Sorites paradox to suggest the possibility that vagueness may be ontologically real; 'solving' the paradox is then missing the point about what the paradox is attempting to demonstrate.

Formally, this is 'solved' by notions such as probabilities, or fuzzy logics; however here a number is definitely assigned to all cases - as in your solution.

A better possibility which retains the sense of the paradox are modal logics that use indeterminate concepts such as possibility - without specifying how possible they are; one might contemplate such a logic where possibilities are ranked by order rather than number - but I don't know if any such work has been done.


It is also called the paradox of the heap.

In essence you did solve the problem, however it is not a Paradox. The problem is deceptive and lies within the weakness of the language itself. It is a misdirection and NOT a paradox. Here are alternate answers I came up with (and I am sure many others have too):

  • The person who asserts the paradox must define a "pile" since it is the person who asserts a position that is required to define the terms adequately for the context of the argument. Therefore you can demand for the assertion that they give you an exact number of grains of sand are in the measurement called a "pile".
  • Since a "pile" is an arbitrary measurement, you can choose to have a pile of 1. Therefore each grain is a "pile" and collectively it is also a "pile" such as a flock of sheep.
  • Instead of 1 grain you can choose N+1 or 2 or more grains are a pile.
  • One can state that the "pile" is a conceptual model so as soon as you alter it in any way it is no longer the same pile and therefore not the same pile.
  • You can also say that the paradox is conceptual, so even 0 grains of sand are still a "pile" that is no longer assembled.
  • The removal of each grain, if each grain is it's own "pile" requires you to put those grains somewhere, since matter can not be destroyed completely, they are somewhere.
  • From a temporal perspective, you can always travel back in time theoretically and the pile would be there, therefore it is always a pile.
    • The concept of the "pile" is not defined and so the question is malformed and meaningless. You may punch the person in the nose that asked the question :D

That's probably enough for now. Enjoy.

PS. you can also have negative amounts of grain as a pile too, but that one is hard to describe.


It's not about the heap or the grains, those are just part of the example given.

The paradox is about vagueness in language.

In the example, "heap" in the sense of "a lot of grains" is vague as it doesn't set exact numbers, because usually heap is not defined as "two or more" or "between 13 and 25000", but just "a lot of". That's fine for most usage scenarios of heap, but not sufficient when it comes to performing arithmetic on its elements. Arithmetic as an "exact" operation requires an exact definition of its predicates. In other words, I can subtract 5 from 279 but I shouldn't attempt to subtract 5 from "a large number" and expect to get a meaningful answer.

I would attempt a simple "proof": When you add something to a large number, you get an even larger number - right?

I) "Large Number" + 5 = "Even larger Number"

II) "Large Number" + 20 = "Even larger Number"

Subtract Equation I from Equation II:

"Large Number" + 20 - ("Large Number" + 5) = "Even larger Number" - "Even larger Number" Result: 15=0 Hence the paradox.

What seems to work fine in language, doesn't work mathematically when you look too closely due to the non-exactness of the components/definitions because the "in-exactness" of the definition can hide a small number or detail.

I can think of a number of real life applications of this:

In computers (or calculators) depending on the actual representation of a number (for example as type "real") it is possible to have an actual number (e.g. 1.2345e765) and add 1. The result is the same as the original number because the representation is too imprecise to cover the full span between 1 (which is 1.0e0) and 1.2345e765. Therefore if I do this: 1.2345e765 + 1.0e0 - 1.2345e765 the result is 0, which is of course mathematically wrong despite the fact that the computer or calculator didn't technically make a mistake. Paradox.

One of my professors (physics, a "large number" of years ago), would harshly reprimand us if we used numbers without units. E.g. "15" instead of "15 km". To him that was rendering the number meaningless and hence wrong even if we had the correct numerical value. NASA I believe ran into that problem when they miscalculated some trajectories wrongly due to mix ups in units, miles/km and such. Specifically this can trigger misunderstandings and errors like "Megawatts" versus "Gigawatts" and the like, where potentially large exponents hide in the units instead of the numerical value. Often entire industries use units to which many members of that industry don't know the exact definition, for example "micron" and "mil".

When Clinton defended himself "I did not have sex with that woman" he artificially interpreted "sex" as purely "intercourse" and therefore the "bj" wouldn't apply. Of course, he kept that definition to himself knowing full well that most of society would interpret "sex" to include "bj". He thus made a claim which he understood to be interpreted by most people as "nothing happened" while in case his true action was proven, he could claim to not having lied (according to HIS definition).

Granted, the last two examples might be viewed a little far-fetched for the question, but my point is that the paradox in question triggers when "details hidden by specific vagueness of predicates" become relevant.

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