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.