In one of Raymond Smullyan's (quite excellent) puzzle books, he offers the following proof that everyone is either conceited or inconsistent. The argument is quite simple: consider all the things that you believe. Do you think they're all correct? If so, well, you're conceited, because only a conceited person would think they're never wrong. But if you don't then you believe something and also believe it's wrong, which is inconsistent.
(Smullyan is an actual math professor in logic, so it's not surprising that this proof is related to a real mathematical phenomenon known as omega-inconsistency.)
I was thinking about this paradox because of an episode of Law and Order: SVU which I saw a few weeks ago. The show centers on a doctor who denies that HIV causes AIDS, and some of his patients, and the show unambiguously condemns the doctor. Strikingly, despite the fact that it's fairly clear that the doctor genuinely believes in what he's saying, he is treated more like a willful criminal than someone dangerously mistaken. Indeed, in at least one point, a police officer suggests (without evidence) that his views must be maliciously negligent, in a manner that suggests the viewers are expected to agree.
But Smullyan's paradox is in play here. We believe, with justification, that the doctor is wrong in this case, and causing harm to his patients. But, somewhere, there's probably another doctor refusing to give a treatment, and that doctor happens to be making the right choice, and will someday be feted, while their persecutors will one day be condemned as cruel agents of an inflexible and indifferent establishment.
I don't have a particular brilliant summing up to give here. My point is just that there doesn't seem to be much general thought about how to strike this balance. When the issue comes up, the response seems to be dominated by emotional disgust, mixed with an abstract idea that there are some limits, but no effort to reason to an actual decision that's independent of the particular case. (Strikingly, the SVU episode dealt outcomes to characters pretty much in linear order of how sympathetic the character was.)
Sunday, December 14, 2008
Wednesday, November 26, 2008
Less unique thoughts on the subject of more unique
On my mind is the sometimes controversy about the phrase "more unique" and its kin. Various authorities claim that the phrase is illogical or ungrammatical.
The argument goes, more or less, that since unique means "Being the only one of its kind", and, which is an absolute property, it makes no sense to say that something is "very unique."
My problem with this argument is that it's bad mathematics, or at least, bad philosophy of mathematics. The issue is that saying something is "the only one of its kind" assumes a well defined notion of what a "kind" is. We're misled by this because usually it is clear what the appropriate "kind" is, and this makes it easy to forget that in some cases there may be multiple reasonable choices.
Every object has some list of properties: just a list of things true about that object. Presumably two objects have the same kind if they have some combination of properties in common. But which properties need to be shared before two objects are the same kind?
Certainly two objects can have some differences while still being the same kind. For example, the desk I'm sitting it is in a different location from any of the thousands of others of the same model, but surely that's not enough to be unique. Even if we ignore the issue of location, this table has different scratches, a slightly different age, and such. So two objects can have a long list of differences and still have the same kind.
But now we make an argument as follows. Suppose that I have some object, let's call it x, and some property P (which may be the combination of a long list of simpler properties), and we assert that the kind of x consists of exactly those objects which have property P. That is, x is unique just if it is the only object with the property P.
The problem is that for any mundane example of a unique object x, the property P is impossible to pin down. Let's take an example. What does it take for a table to be unique? Most people would find it reasonable to say call a table unique if it is made as a project by an artist who sculpts each piece by hand, and makes only one such table. What if the artist makes three tables, each with different designs, so no two are identical; can all three tables be said to be unique? If so, what if we then outsource to a computer program which, via some elaborate algorithm, carves ten thousand tables, each with a different randomized pattern, though it might take careful examination to find the differences.
If these ten thousand tables are all unique, we ask the question again with the tables more similar, and then more similar still, until they are identical. There's no point along this process which is absolute, so if uniqueness is an absolute concept, it must be that making the second table already made the first one non-unique. That is, the property P for this table must include the fact that it is the only one made by this artist.
Surely, though, other artists have made tables of their own. Suppose a second artist comes along and is inspired to make a table of their own in similar, but not identical, style. Then a third artist, and a fourth, until soon we have ten thousand tables, all rather similar, but no two exactly the same. If these are all unique, what if the styles are even more similar? Or nearly indistinguishable? Somewhere along this line, the tables stop being unique, and if the demand is that uniqueness be absolute, there's only one stopping point: as soon as there's a second table which is even similar, both tables are not-unique.
Undoubtedly, though, some other artist, somewhere, at some time, has also made a handcrafted table. So we play the same game again: after ten million artists each independently handcraft a table, are they all still unique?
Any property P that an object in the mundane world might have faces this problem. The real world isn't one of perfect absolutes, so there will always be the possibility of objects which are "almost P." If these objects are all unique, we ask about objects which are "very almost P", and so on; since we are demanding an absolute notion, there's no line to be drawn, until we reach objects that are extremely similar to P, similar enough to violate any non-trivial notion of uniqueness. On the other hand, if these objects aren't all unique, we make them less similar, until they're really not similar at all. Again, the only absolute line is at the extreme, so the notion of uniqueness becomes so rarefied as to be useless, because it transcends almost anything we could every experience---a term that applies only to a handful of objects, perhaps the universe itself, whatever divinities you believe in, and not much else.
(To anyone who responds that this is as it should be, my answer is that neither history nor current usage stands with you. Perhaps you wish to change the language by imposing a new meaning on the word, but you are no longer discussing any sort of objective notion of the language.)
This entire problem is easily resolved by surrendering the notion that uniqueness must be absolute. The "kind" of an object---the choice of the property P---can be context dependent. This is not, as the sources above claim, illogical. Indeed, this is the same approach standardly taken in rigorous mathematics. Notions like "identical" and "same" are fraught with paradoxes (Wikipedia has decent entries on the big two, the Ship of Theseus paradox and the Sorites paradox), so in mathematics, these notions are shorthand for something like "sharing all properties relevant to the current discussion."
Once we accept that "unique" has this hidden, somewhat fuzzy assumption, terms like "more unique" become clear and straightforward. Like most things in real language, there's some room for ambiguity, but in practice it's generally pretty clear. One object is more unique than another if it is unique relative to a stricter property (where the notion of being a stricter property may itself be rather vague). Indeed, when terms like "more unique" or "very unique" appear in casual use, this turns out to reliably explain their meaning.
(Some people object to "more unique" on aesthetic, pragmatic, or ideological grounds, for instance arguing that it contributes to weakening the word "unique," or is frequently used in ambiguous ways, or that it's used often and vacuously in commercial advertising. This has nothing to do with whether the usage is logical, which is the only thing I addressed above, and perhaps there are persuasive arguments along these lines that the usage should be avoided. These are ultimately subjective arguments, though, and need to be defended in the murkier waters of opinion, rather than cloaked by incorrect arguments on objective grounds.)
The argument goes, more or less, that since unique means "Being the only one of its kind", and, which is an absolute property, it makes no sense to say that something is "very unique."
My problem with this argument is that it's bad mathematics, or at least, bad philosophy of mathematics. The issue is that saying something is "the only one of its kind" assumes a well defined notion of what a "kind" is. We're misled by this because usually it is clear what the appropriate "kind" is, and this makes it easy to forget that in some cases there may be multiple reasonable choices.
Every object has some list of properties: just a list of things true about that object. Presumably two objects have the same kind if they have some combination of properties in common. But which properties need to be shared before two objects are the same kind?
Certainly two objects can have some differences while still being the same kind. For example, the desk I'm sitting it is in a different location from any of the thousands of others of the same model, but surely that's not enough to be unique. Even if we ignore the issue of location, this table has different scratches, a slightly different age, and such. So two objects can have a long list of differences and still have the same kind.
But now we make an argument as follows. Suppose that I have some object, let's call it x, and some property P (which may be the combination of a long list of simpler properties), and we assert that the kind of x consists of exactly those objects which have property P. That is, x is unique just if it is the only object with the property P.
The problem is that for any mundane example of a unique object x, the property P is impossible to pin down. Let's take an example. What does it take for a table to be unique? Most people would find it reasonable to say call a table unique if it is made as a project by an artist who sculpts each piece by hand, and makes only one such table. What if the artist makes three tables, each with different designs, so no two are identical; can all three tables be said to be unique? If so, what if we then outsource to a computer program which, via some elaborate algorithm, carves ten thousand tables, each with a different randomized pattern, though it might take careful examination to find the differences.
If these ten thousand tables are all unique, we ask the question again with the tables more similar, and then more similar still, until they are identical. There's no point along this process which is absolute, so if uniqueness is an absolute concept, it must be that making the second table already made the first one non-unique. That is, the property P for this table must include the fact that it is the only one made by this artist.
Surely, though, other artists have made tables of their own. Suppose a second artist comes along and is inspired to make a table of their own in similar, but not identical, style. Then a third artist, and a fourth, until soon we have ten thousand tables, all rather similar, but no two exactly the same. If these are all unique, what if the styles are even more similar? Or nearly indistinguishable? Somewhere along this line, the tables stop being unique, and if the demand is that uniqueness be absolute, there's only one stopping point: as soon as there's a second table which is even similar, both tables are not-unique.
Undoubtedly, though, some other artist, somewhere, at some time, has also made a handcrafted table. So we play the same game again: after ten million artists each independently handcraft a table, are they all still unique?
Any property P that an object in the mundane world might have faces this problem. The real world isn't one of perfect absolutes, so there will always be the possibility of objects which are "almost P." If these objects are all unique, we ask about objects which are "very almost P", and so on; since we are demanding an absolute notion, there's no line to be drawn, until we reach objects that are extremely similar to P, similar enough to violate any non-trivial notion of uniqueness. On the other hand, if these objects aren't all unique, we make them less similar, until they're really not similar at all. Again, the only absolute line is at the extreme, so the notion of uniqueness becomes so rarefied as to be useless, because it transcends almost anything we could every experience---a term that applies only to a handful of objects, perhaps the universe itself, whatever divinities you believe in, and not much else.
(To anyone who responds that this is as it should be, my answer is that neither history nor current usage stands with you. Perhaps you wish to change the language by imposing a new meaning on the word, but you are no longer discussing any sort of objective notion of the language.)
This entire problem is easily resolved by surrendering the notion that uniqueness must be absolute. The "kind" of an object---the choice of the property P---can be context dependent. This is not, as the sources above claim, illogical. Indeed, this is the same approach standardly taken in rigorous mathematics. Notions like "identical" and "same" are fraught with paradoxes (Wikipedia has decent entries on the big two, the Ship of Theseus paradox and the Sorites paradox), so in mathematics, these notions are shorthand for something like "sharing all properties relevant to the current discussion."
Once we accept that "unique" has this hidden, somewhat fuzzy assumption, terms like "more unique" become clear and straightforward. Like most things in real language, there's some room for ambiguity, but in practice it's generally pretty clear. One object is more unique than another if it is unique relative to a stricter property (where the notion of being a stricter property may itself be rather vague). Indeed, when terms like "more unique" or "very unique" appear in casual use, this turns out to reliably explain their meaning.
(Some people object to "more unique" on aesthetic, pragmatic, or ideological grounds, for instance arguing that it contributes to weakening the word "unique," or is frequently used in ambiguous ways, or that it's used often and vacuously in commercial advertising. This has nothing to do with whether the usage is logical, which is the only thing I addressed above, and perhaps there are persuasive arguments along these lines that the usage should be avoided. These are ultimately subjective arguments, though, and need to be defended in the murkier waters of opinion, rather than cloaked by incorrect arguments on objective grounds.)
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