The Sorites Paradox Isn’t

Clearly, a guy with no hair on his head is bald. But so is a guy with just one—if and only if we define bald as “a man with little or no hair.” If the guy has one hair and we define bald to mean “a man with no hair” then the man with one hair is not bald. So let us use “a man with little or no hair” as our definition and see where that gets us.

We assume that if a man with one hair is bald (by our definition), then so is a man with just two hairs. And if a man with two hairs is bald, then so is a man with three. We can expand this: if a man has N hairs and is bald, then a man with N + 1 hairs is also bald. Thus (eventually) a man with a million (say) hairs on his head bald, too. Which is absurd. Any man which such a mane is clearly fully flocked. Yet our derivation is error free.

This is the Sorites, an ancient puzzle, also given with respect to grains and heaps of sand (the words is derived from the Greek heaped up). More than a few writers on this paradox, after reaching the gotcha!, now say something like the following:

“We seem to have reached the point where we say that a man with, say, 5,000 hairs is ‘bald’, but one with just one more tiny, wee hair is not. This is nuts. Nobody can see the difference between 5,000 and 5,001 hairs. Something must be wrong with our system of logic.”

The man who says this, or anything like it, makes (at least) two mistakes. I’ve already given a hint of the first error above. There is nothing wrong with logic, but there is with the definition of bald. That word, when used in this exceedingly formal logical argument itself becomes a formal creature. It is no longer the bald as used colloquially, it is instead like the X used in algebra. It is an abstract thing, it no longer means real baldness on real men. It means logical X-ness on fictional men.

Indeed, rewrite the Sorites to remove the pseudo-word bald and replace it with X. X now means a man with fewer than Y hairs. If the man with no hairs is X, then so is the man with one hair, and so forth. Now, at some point we either bump up against Y, in which case the man is no longer X, or Y is the limit and the man is always X except at the limit.

If I were to have originally written the Sorites in this algebraic form—with just Xs and Ys—there never would have been a gotcha!, we never would have questioned the foundations of logic, there would have been no paradox. That there felt like one when we do use bald instead of X can only mean that we are silently augmenting our argument with hidden premises (which define bald). We figure that because these premises are unstated, or do not appear in print, they are not truly there.

One hidden premise is that the word bald to me, and to me right now, means a man with a certain shape of head and a certain lack of hair. I need not know how many hairs this man has, but I will make the judgment bald or not by what I see. Of course, we may, after my judgment, count the man’s hair and thus reach a quantification. My premises fluctuate: they are different for different times and men, or for the same men but they change depending on what these men wear, or the properties of the light, my relations to these men, or even by how much I have drunk.

My premises are almost certainly different than yours. I may say bald when you do not. That our behavior is not constant or that our judgments do not agree is meaningless. Neither is it relevant—and here is the second mistake—that I cannot articulate my premises. All that I can do is to say bald or not. Quantification, as I said, can always be had after the fact. But all this will tell us, in any individual case, is that the man now in front of me has not yet reached Y, or that he has exceeded it. We will not be able to deduce Y (unless the man is willing to undergo experimentation; however, my premises might change as we add or subtract hair from our recruit).

Unacknowledged, hidden premises are the generator of many “paradoxes.” The most relevant to statistics are in (faulty) criticisms of Laplace’s Rule of Succession, which we can attack another day.

28 Comments

  1. I think there’s also something to be said about creating dichotomies when in fact a continuum exists. It’s much easier for us to classify people as bald or not, but certainly we can understand that a scale of baldness exists. A man with 5,000 hairs is less bald than a man with 4,000 hairs who is less bald than a man with 3,000 hairs, etc.

    So the question shouldn’t be: are you bald or not?

    Perhaps the question is: where on the baldness scale do you lie?

  2. Unacknowledged, hidden premises are the generator of many “paradoxes.” The most relevant to statistics are in (faulty) criticisms of Laplace’s Rule of Succession, which we can attack another day.

    Oh come on, what a tease!

  3. Though it’s not completely wrong to say that the logic is a problem. Forcing judgment-related concepts like “baldness” into binary logic is pretty limiting. At least consider File Not Found, for crying out loud.

    Seriously, though, this is where fuzzy logic and fuzzy numbers make more sense as far as quantification go. Though an easier example than “bald” is probably “tall.” How else could you attempt to quantify something like:

    Q: “Is he tall?”
    A: “Well, sort of. I mean, I wouldn’t say that he’s short, but…”

    Either way you want to look at it, it’s still just another example of not understanding the assumptions you’re making when looking at an argument, as you say.

  4. Some similar “paradoxes”:

    How many cars must one own before one is “rich”?
    How long must one go without food before one is “starving”?
    How long must one speak a language before one is “fluent”?

    Obviously, the answer to all of them is “it depends” (or perhaps “what a stupid question”), and so it is with baldness and the number of hairs.

  5. Yesterday at Judith Curry’s blog a commentor found no fault with the hypothesis: The Global Surface (air) temperature will rise about 0.2 degrees per decade.

    About?

  6. Logic might be infallible in its own realm, but it is a realm only connected to the real world through the experiences of us, the creatures capable of logic.

    Our interaction with the world is not mediated through the kind of logic we use to solve theoretical problems. We humans are great at making fast, holistic judgements. This is something completely different from logic. Logic is a powerful tool that we can use to think about stuff, but logic in itself requires no thinking (any computer performs logic much better than people ever will).

    What you call hidden premises is connected to us humans being in a context. There’s a world surrounding us and even while most of what’s going on in it is inaccessible to us we exist in an ever changing reality. This is natural to us, but reality is of course way too much to include in any thought, sentence or calculation. Therefore we use models of reality that we express in a concise fashion. Sometimes these models give weird results, that is just a consequense of the incompleteness inherent in any model. Just as words need to be embedded in a language to be meaningful, so does models need to be embedded in a network of models to be meaningful. Just as there are different languages there are different ways of understanding the world, or different “premises” that guide our thoughts.

    It is helpful to identify important premises, it’s the way towards a shared understanding of the world. However, there will always exist premises that are not shared, due to the simple fact that no two persons are the same.

  7. The paradox arises from confusing quantity with quality which is what fuzzy numbers are supposed to avoid. Quality is not a hidden premise — it’s how people operate without the necessity to count fingers and toes for every operation — helps when being chased by a tiger. A better definition of bald is: hair covering far less area than normally expected. Of course, you are the stuck with quantifying “far less” and “normally expected” both of which are ultimately a judgment calls. Maybe it’s safer to define bald as anyone without a hat.

  8. Tom M/All,

    Your final question is fine and can be answered, but this is not the difficulty with the Sorites. We are not interested in baldness per se, nor in heaps of sand, just as we don’t care about Socrates’s mortality in the classic syllogism. We want to know why the structure of this kind of logical argument appears to end in paradox, whether it be used for baldness, heaps, whatever.

    We also do not have a continuum. We are working with hair at the quantum level; which is to say, the discrete level. We can only have quantum hairs, quantum grains of sand and so on.

  9. Sometimes even the spoken assumptions can get one in trouble, if chosen poorly. Baldness being determined by counting the number of hairs on the head, the head including the face, the “cure” for baldness can be found in growing a beard. Even were a well bearded man to have no hair on his skull, he would still have enough hair on his head to avoid meeting the numerical definition of baldness.

  10. Adde parvum parvo magnus acervus erit – Ovid 43 BCE – 17 CE

    “Add little to little and there will be a big pile”

  11. Briggs said: “we can only have quantum hairs, quantum grains of sand and so on”.

    This clearly calls for a quantum theory of baldness.

    Axiom: The number of hairs in each follicle of the scalp is either one or zero.

    Baldness can then be defined as the ratio of number of follicles with zero hairs to the total number of follicles.

    This is not a continuous function, but could be mistaken for one from a distance (say 10 feet or more, depending on eyesight).

  12. Surely baldness is a relative measure of how much skin is visible in places where there is normally hair. ie we measure skin area rather than hairs

    And relative because, as you say, there is a grey area where we disagree about a particular person’s baldness.

  13. Bah — this is baby stuff. If you would only read “Science and Sanity” by Count Korzybski then all would be clear……..hair #1, hair #2, hair #3……………

  14. William
    .
    We want to know why the structure of this kind of logical argument appears to end in paradox, whether it be used for baldness, heaps, whatever.
    .
    Because it is only disguised as a paradox . Logics is not at fault here .
    The only operational definition of logics I admit is formal logics .
    Among possible choices , our human brain has a strong preference for the Peano variety .
    It is probably because the formal logics governing the validity of statements elaborated by our brain is emulating/simulating the physical processes going on in the brain itself .
    Then the evolution selected the brain processes expressed f.ex by “P or non P is always true” because this process is clearly superior for the survival to a brain process expressed by “P or non P is sometimes false” .
    In the former case I know that this striped thing in front of me is or is not a tiger and will act accordingly while in the latter case I risk to get eaten by doing nothing .
    I suspect that a cat’s brain also works on the mode “P or non P is always true” even if it cannot (yet) write a paper about it .
    .
    So this “paradox” is not really a paradox because it starts with an undefined term and then during the argument stealthily implies a definition .
    It is equivalent to a statement “A segment is short if its length is small” .
    Even if this is a continuous (non discrete) case , it allows to infer that all segments but the one of infinite length are short .
    This is also not a paradox it is just a possible (arbitrary) qualification of all finite length segments .
    Not a very useful one at that because the set of finite length segments and the set of short segments would be identical , so the statement just defines a synonymie .
    .
    In your initial statement “bald” and “having hairs” are operationally equivalent (synonymes) because “little” is undefined and therefore logically irrelevant .
    So it is not surprising and perfectly valid in formal logics that you infer that “Anybody who has hairs is bald .”
    Just define correctly “little” and the “paradox” disappears .

  15. RWS,

    Have you a link for us?

    An Amazon reviewer to the good Count’s book said, “The reason for this is that Korzybski summarized his system as a non-verbal diagram. Probably, the wisdom of thousands of books are represented non-verbally on that diagram!”. All of the world’s knowledge in just one diagram, eh? I can’t wait to see it.

    TomVonk,

    All true, but what upsets classical authors is that “just one hair” or “just one grain of sand” demarcates bald/not bald and not heap/heap. “How can just one hair make a difference?” is their question. Well, by the definition of the argument, supplied with hidden premises on the definition of bald/heap, it just does.

  16. I fear the good Count isn’t as famous as he used to be, if he were I suppose you would have recognized my post for a rather lame version of what it was. Your best source for Korzybski is probably “Fads and Fallacies in the Name of Science” by Martin Gardner. You’ll save a lot of time that way………..

    He did obsess about this sort of thing, though……..

  17. RWS,

    Aha! That’s where I recognize the name. I’m away from my bookshelf at the moment, but I can’t wait to get back and re-read Gardner.

  18. Correlary story —

    If I earn a dollar I am not rich. If I earn 1,000 dollars, I am still not rich. If I had $100,000 dollars in my bank account, some people would say that I am rich. But, I would feel a long way short.

    There is always some threashold in my mind where I would call myself rich. But, as I accumulate wealth, my definition of rich keeps moving. In fact, it moves faster than my net worth. I now realize that even if I had $10 million, I would still be short.

    Maybe at $100 million…

  19. I don’t think there’s any need of talking about hidden premises. There’s simply no reason to believe that what people mean by “bald” is “having less than or equal [N] hairs on the head.”

  20. Doug M There is always some threashold in my mind where I would call myself rich. But, as I accumulate wealth, my definition of rich keeps moving. In fact, it moves faster than my net worth. I now realize that even if I had $10 million, I would still be short.

    Maybe at $100 million.

    How would your definition stop moving at $100 million? If you think it would, this means that you define “rich” as “having a net worth of at least $100 million.” This is not consistent with the claim that it keeps moving.

  21. Doug M;

    If I earn a dollar I am not rich. If I earn 1,000 dollars, I am still not rich. If I had $100,000 dollars in my bank account, some people would say that I am rich. But, I would feel a long way short.

    There is always some threashold in my mind where I would call myself rich. But, as I accumulate wealth, my definition of rich keeps moving. In fact, it moves faster than my net worth. I now realize that even if I had $10 million, I would still be short.

    Maybe at $100 million…

    Your problem is trying to take a quantitative measure of a qualitative attribute. One is rich when one has more wealth than is sufficient to meet one’s needs. Britton’s Second Law (bills will always expand to meet the available cash) teaches us that one can only be rich for short periods when wealth has increased and the bills generated by this new wealth are still in the post.

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