If a bald man grows just one hair on his head, he is still bald, right? If a billionaire gives away a dollar to a poor man, he's still rich. Even if he does it again, he's still rich and the poor man is still poor. A heap of sand may have a million grains of sand it in. Remove one, and it's still a heap (or in Greek, sōros). Remove another grain, and another, and it's still a heap. Is it still a heap when there are ten grains left. How about one? None? When does the heap cease to exist?

As is freely admitted, these are not precise categories. Therefore they will not have precise boundaries. One must accept that the truthfulness of P is not a yes or no thing. As N increases, then the truthfulness of P decreases and the truthfulness of not-P increases.

This means, for instance, that baldness decreases with increasing hair. A man with two hairs is very slightly less bald than man with one hair. By the time you get to a man with a hundred thousand hairs, he is not very bald at all. In fact he may be more not-bald than he is bald, and be somewhat hairy.

The sorites paradox is just one of a class of paradoxical arguments that raise difficult questions about logic, language, and most importantly, vagueness. We see the problem when we try to formalize the semantics of vague terms. The word 'sorites' derives from the Greek word soros meaning 'heap'. In its most famous form, the paradox is expressed as follows:

(1) 1 grain of sand is not a heap
(2) For any n grains of sand that does not constitute a heap, n+1 grains does not make a heap either
(3) Therefore, 7 billion grains of sand does not constitute a heap

Implicit and irrelevant in this argument is the assumption that the grains of sand are arranged in a manner conducive to maximal heap height (they are not spread out in one layer, they are piled, etc)

We see here the difficulties that arise when one considers a problem such as this. Well-nigh unobjectionable reasoning seems to have led us from well-nigh unobjectionable premises to an obviously absurd conclusion! Something is wrong here, but it's not clear what!

To rexamine the situation, our logic is obviously correct. In the example above, we are using nothing more than the Modus Ponens and Universal Instantiation to connect (1) and (2), but the problem could just as easily be formulated to only require MP. This is just about as trivial as logical reasoning can get.1

So the logic is correct, that must mean that our premises are incorrect. Yet, our premises seem perfectly secure: obviously one grain of sand is not a heap (1), and secondly, if I have a certain number of grains of sand and they are not a heap, one more grain of sand is not going to make the difference (2). If our premises are incorrect, then perhaps we should assume their opposite. The negation of (1) would be that one grain of sand does in fact constitute a heap. This seems absurd, and against what we commonly take 'heap' to mean. The second thing we could modify would be to take (2)'s negation. But this seems equally, if not more absurd. If we were to assume its negation, it would mean that somewhere along the line, there is a number n that does not equal a heap, but n+1 grains of sand does. This is simply absurd. How could one grain of sand, (most likely an imperceptible addition), make the difference? And if so, what is that n? Is it 5,677,801? 2,328,190? Where?

This is obviously a connundrum because if our premises seem to be correct, and our reasoning does too, what's left for there to be wrong? The Sorites Paradox is one that has successfully escaped being solved by other forms of logic and may continue to be more and more of a problem as we try to get machines to understand human language. The problem is in fact bigger than merely looking at what constitutes a heap--vagueness is everywhere in our normal language. (For a better discussion of this, please see Vagueness). What does it take to create a logic of vagueness? Can it ever be useful for language to contain vague predicates? What is wanted is a compelling resolution of the paradox, one that not only blocks the absurd conclusion, but what's more does so in a manner that illuminates the use of vague language generally and helps explain why the argument, though it must be unsound, nonetheless seems to consist of true premises and valid reasoning.

More Examples of the Sorites Paradox

  • The example of baldness. Just how many hairs does one have to have in order for the term "bald" not to apply? Is there a sharp boundary between being bald not not bald? If so, how many hairs makes this distinction? One objection to this is that bald = hairs, and therefore it is very easy to make the distinction. This however doesn't address the fact that in everyday life, people use the term bald to refer to people with some hair (e.g. Patrick Stewart). It is much more interesting to try to account for this in some sort of logic then to merely dismiss it as a semantic issue.

  • The reverse of the heap analogy. We can also look at the heap problem from the reverse standpoint.

    (1) 7 billion grains of sand constitutes a heap
    (2) For any n grains of sand that constitutes a heap, n-1 will also be a heap
    (3) Therefore, 0 grains of sand consitutes a heap

  • Color chips. Say I had a jar full of color chips that went from orange to red in imperceptible steps (two adjacent chips look identical to each other). If I tell you to pull out a red chip, you'd have no problem doing so, but if I were to ask you to sort them, you'd have a problem. Additionally, if I were to ask you if there were an odd number of red chips, you would be unsure of what to say.

1It should also be noted that by specifying a specific number of grains for (3), we avoid invoking the Principle of Mathematical Induction, which might possibly raise some concerns. In its simplest form this method of induction takes a case n, shows it to be true for case n+1, and then assumes it for all possible n. In this example, we aren't saying anything about all possible n, merely 7 billion.

I am grateful to Steven Gross, philosophy professor at the University of Pennsylvania whose lectures and article An Invitation to Vagueness have been an invaluable resource on this subject.

Two new thoughts on the paradox. JerboaKolinowski is entirely correct when he says that the problem stems from the ambiguity inherent in the term 'heap'. It is a fundamentally vague predicate, just like "young" or "middle aged". What we have here is what we witness in the formation of any semantic category: prototype effects. In any attempt to define something, we do not have a set of necessary and sufficient criteria which allow us to uniquely specify the meaning of the word. What tends to happen is that we possess certain prototypes which are at the center of our categories (apple is a prototypical fruit) and everything else has varying degrees of fitness. In the sorites context, there is no fixed definition of 'heap', merely that we have protoypical notions. This means that any formal, specific attempt to specify how many grains of sand equal a heap will inevitably fail. (This is like starting with a whole apple and saying "Does this equal an apple?" "How about if I take a bite?" "What about two bites?")

The second response was offered by Dr. Murray Grossman, a neurologist I performed research with. He suggested the problem lies in the fact that "heap" is a mass noun, like "sand", and you can't specify numbers with mass nouns. Therefore, any attempt to "count up to" a heap will fail -- you can't make the switch from count to mass.

To my mind, this paradox best serves to illustrate the perils of assuming a straightforward relation between natural language and logic.

The fact is (as pointed out in the formal discussion of the Sorites paradox) that the term heap does not unequivocally draw a clear boundary line, separating one set of objects (all of which are heaps) from everything else (which are non-heaps).

As there are no rigidly defined criteria for saying whether something is a heap or not, our best evaluative technique is to ask competent users of English whether the word is applicable in a given case (one of whom may be ourselves). Clearly, there are some collections of grains of sand we can unhesitatingly identify as heaps, and some as non-heaps. Between these, we may scratch our head for a while, and then issue a verdict (or admit ignorance). But another person, faced with the same decision, may go the other way. We ourselves, for a particular instance, may make the opposite decision at a different time. So we are looking at a statistical measure (how many people at a given time will say "heap") in these problem cases, at best.

Suppose we actually conducted an experiment - take 100 people, and some sand. One by one we add grains of sand to our collection, and at each turn we ask each of the 100 people to issue their verdict (keeping them from chatting to each other). We might tabulate the results, as follows (and I admit I'm guessing the results, not having done the experiment!):

number of grains       heap        non-heap    don't know
        1              0%            100%         0%
        2              0%            100%         0%
        3              0%            100%         0%
(everybody agrees these are not heaps!)
      m - 1            0%            99%          1%
      m                1%            99%          0%
      m + 1            2%            98%          0%
      m + 2            1%            99%          0%
      m + 3            1%            98%          1%
      m + 4            1%            99%          0%
(one of the subjects has thought about it, and decided that a really small heap counts as a heap, another subject considers the question, wavers, and then decides for the majority view)
      n               51%            34%          15%
      n + 1           50%            36%          14%
      n + 2           51%            29%          20%
(looking more like a heap ... n is the first number for which the majority of our subjects say "heap".)
      h - 1            99%           0&            1%
      h               100%           0%            0%
(h is the first number at which all subjects agree on "heap")
      h + 5            99%           0%            1%
      h + 6            99%           1%            0%
(It could happen!)
      i                100%          0%            0%
      i + 1            100%          0%            0%
      i + ...          100%          0%            0%
(Beyond i, there are no further waverings, and everyone agrees, as long as we conduct the experiment, that "heap" is correct - of course if we continued the experiment indefinitely, we'd find people thinking "that's not a heap, that's a mountain" (or a planet ...) and so the numbers would go down again. But never mind that!)

Well, it's an imaginary experiment, but I don't find the results implausible - I'm inferring and extrapolating them from my own experiences of the trickiness of certain aspects of language, and my observations of others. (Although the experiment is perhaps flawed - in real life, people do talk about the terms they use in language, and strive to agree about them. Also, the effect of all the heap-decisions on the behaviour (and sanity!) of the experimental subjects is ignored, though we could address this, perhaps, by getting a different group of 100 people to cast their verdict for each additional grain.)

We can see from these considerations, I hope, that there are at least two errors in the statement of the paradox. The first is the unspoken assumption that for any object and any language-user it is a case of either a strict heap, or a strict non-heap - this manifests in step 2, which might be true as far as it goes, but says nothing about objects which are neither (strict) heaps nor (strict) non-heaps. It is only in logic that the law of the excluded middle holds without fail - in real life, we may sometimes be unable to reach a decision.

The second flaw follows: in practice, in the case of an individual language user, there will indeed be some number (m, in the case of our first heap-sayer, above) where m grains strikes a person as a heap, and m - 1 does not. The number will differ for different people, and it may be that m + 1 will not strike the same person as a heap - because we simply do not work out whether something is a heap by counting the grains! We look it over and think "heap" or not, as the case may be.

Ok, we might then go on to say: "So which is the correct number, for which we should say that all heap-shaped collections of grains of sand, with that number of grains, is really a heap? Is it m, n, h, or i?"?

And here is the root of the problem. The paradox is taking the word 'heap' as representing an abstract reality which our real-world examples either conform to or don't, and therefore should properly be called heaps or not, as the case may be - which is to assume that there is some number before which is is simply incorrect to say "heap", and beyond which it is incorrect not to. Plainly, there is not such a number (bear in mind that the exact values of n, h, and i (which may seem plausible candidates) are all contingent - and will vary according to on which group of people, and on which day, we conduct the experiment).

This idea of a 'god-given' heapness is really just a Platonist folly stuck in the back of our understanding of language (Platonism may be taken more seriously where the objects of discourse are (or appear to be) explicitly defined at all points, like in mathematics - but natural language talk of heaps is not mathematics!) It's more accurate to view this kind of unequivocal clarity as a sort of aspiration of language.

If we accept that there is no universal god-given category of heapness against which all collections of grains of sand must be measured, things begin to make a little more sense.

Ludwig Wittgenstein used the phrase "penumbra of meaning" to point out this property of language - that though there are dark areas which are fully inside the denotation (shadow) of a term, and light areas which are fully outside it, there may well be areas which belong in a fuzzy border region (the penumbra).

Does this mean we can never correct anyone in their use of the word heap? No. Clearly, if someone calls 2 grains of sand a heap, they are wrong, and need correction, just as if they had called an elephant a heap. (In the one case we'd say "no, heaps are bigger than that", in the other case we'd say "not everything that's big is a heap!")

This kind of limited vagueness can sometimes be tremendously useful in language, as in another example, from Wittgenstein's Philosophical Investigations: if I say "stand roughly there" (pointing) you will likely know exactly what I mean!

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