Matt Yglesias goes into philosophy major mode to discuss the Sorites Paradox: a class of arguments generated from vague predicates. For example, if you have one grain of sand, that's not a heap. If you have two grains of sand, that's not a heap either. But if you have ten million grains, that is certainly -- by any commonsense definition -- a heap. But the conclusion of the Sorites paradox denies that ten million grains make a heap (by inference from the fact that an individual grain added does not make a prior non-heap collection of sand into a heap).
So at what point can you say that you have a "heap" of sand? The philosophical problem is that of vagueness, something which philosophers are congenitally given to hate. The problem is applicable to much more than sand heaps: when can you say a man is bald? (That's Yglesias's quandary.) Or when can you say that a human life begins? Or how can you know when someone is rich?
This site details the 4 types of responses to the paradox in formal logic terms. One can (1) take the Frege/Russell way out, and deny that vague expressions have any place in logic whatsoever. Or one can (2) take an epistemically skeptical point of view, and say that the collection of grains becomes a heap, but only at an unknowable, indeterminate number of grains. Or (3) you can domesticate vagueness into a formal system that has more than two truth states (a many-valued logic). Or (4) you can accept the paradox, and conclude that there are no heaps of sand (and conversely, that one grain of sand is a heap).
I think I like Wittgenstein's solution to the problem the best. It's a modulation of the Russell tactic -- ie. he denies that soritical expressions are welcome within the realm of logic -- but it's more subtle, because Wittgenstein's philosophical project embraces much more than Russell's pure formal system of logic. It's quoted on this comprehensive vagueness site:
For remember that in general we don't use language according to strict rules - it hasn't been taught us by means of strict rules, either. We, in our discussions on the other hand, constantly compare language with a calculus proceeding according to exact rules.
"In our discussions," ie., in philosophical circles, we try to apply a calculus of exact rules. But that's exactly what you shouldn't do, according to Wittgenstein, because it obscures the simple truth in everyday language: that heaps of sand exist, and we know them when we see them. The "precise point at which a collection becomes a heap" is just a wild goose chase beyond the boundaries of formal language -- "whereof one cannot speak, thereof one must be silent."
And this is why Wittgenstein became my man in college.
Further: Why do none of these "solutions" address proportionality? If my hair is 1" long and I have a half inch cut off, I've lost a lot of har. But if my hair is 1' long and I cut off a half inch, most people won't notice.
Although this may be just another way of phrasing the problem, not a meaningful solution or addresss.
Posted by: Scot Hacker at March 1, 2004 12:04 AMProportionality is a type of vagueness, yes. And so are many other things, as becomes evident if you look at this bibliography on the subject:
http://www.btinternet.com/~justin.needle/bib_alpha.htm
It's amazing how much verbiage gets produced because philosophers just don't like it when anything is vague or imprecise.
Proportionality is not a type of vagueness when it can be quantified. Before their use in formal systems, the terms must always be defined rigorously. In Mathematics, you cannot even assume a consensus on the meaning of "multiplication". You have to define it. Witness in the mathematics of vectors, that there are two kinds of multiplication (the "dot product" and "cross product") which are distinct. However, they remain powerful tools because as defined they produce homologous meaningful results on very different kinds of vectors (e.g., matrices, polynomials, even linear representations of curl).
Posted by: Andrew Turner at March 1, 2004 12:36 PMAndrew, you're right to say proportionality isn't vagueness if its terms have been formally defined. But that's just the issue about vagueness -- certain types of it (proportionality not necessarily one of them) are not amenable to formalization. If you try to expand the formal system to include the vagueness, you break other aspects of formalism.
Read this site for more on this:
http://plato.stanford.edu/entries/vagueness/#3
Posted by: chris at March 1, 2004 01:47 PM