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.