June 29, 2005
The Isle Of Cats
Suppose there is, somewhere, a little island where a colony of a few hundred or so cats does dwell. How the cats arrived there in the first place, and how they get food is immaterial.And we're only talking about the cats of the island (the elemts of a set that is in turn a subset of the "All cats in the world" set), no other cats.
About these cats, I can state "All cats on the island are black".
This is a strong statement; it says something precise about all the cats of the island. It is also quite easy to disprove: you just have to find one single cat of any colour except black.
So, I may go for a weaker statement: "The vast majority of cats are black".
This is harder to disprove: first, you and I have to agree on a common definition of "vast majority": it can be 2/3, 3/4 etc. Second, you have to count the cats, or do a statistical study in order to back up your assertion (that would be, black cats do not constitute a vast majority).
I may do a statistical study and say: "At 95% confidence, (90 ± 7)% of cats are black".
You do your study too, and conclude that at 95% confidence only (80 ± 5)% of the cats are black. The confidence intervals overlap (even so slightly), so it's quite hard to say which statistics is right. Not to mention that we could accuse each other of having used a flawed method or an unrepresentative sample of the population.
I may also say "There are no white cats on the island".
This is another strong statement, but it's also negative, and this makes things much more complicated. For example, the only white cat may be hiding while you scout the island back and forth searching for it. So a doubt will always remain: you did not see any white cat because there is none, or because you were unable to spot the few present?
Generally, a strong negative statement is rather dishonest intellectually, because of this unresoluble doubt. Not always, sometimes a negative statement is OK; it depends from the context. For example, "There are no electrons with positive charge" is a honest statement, because those particles with positive charge and mass equal to the electron's mass are called positrons.
And all the above supposing that we both are being honest - or at least honest enough.
We may both play dirty, and things will get much more complicated.
For example, when you produce one grey cat I may insinuate that you did not find it on the island, but brought it from the mainland.
I can refuse to agree on a common definition of "vast majority" and thus make discussions about it meaningless.
I may manipulate my statistics, and/or accuse you to do so - this is different from the case discussed above, because the idea of manipulation implies a malicious will; not good-faith mistakes.
Or, when trying to prove the existence of at least one white cat you may publish a dark, out-of-focus, low- resolution photograpy showing a white indistinct blur, and insist that it's the infamous white cat. But it is impossible to tell exactly what the white blur is - accepting that the picture was indeed taken on the island. But when asked to produce better evidence, you say that it's not possible, the white cat won't show itself again but anyway you believe it exists and I should trust you. But my mate Chris says that he lost his white rabbit on the island, just a few hours before the photo was taken...
Pressed further, you then state that the white cat was kidnapped by the Evil Secret Service after you took the picture, because the son of Big Oil Corporation's CEO likes to have his summer holiday on the island, but he loathes white cats and so the CEO used his political connections to have the few remaining white cats kidnapped and killed by the ESS. And save the white cats from the capitalist oppression, too. (It does not make sense, but it isn't very different from popular conspiracy theories).
This is just a fanciful and at times intentionally silly example, but it is based on the rules of logic and, peripherally, set theory. And if you look closely, you may find that it bears some relation with real events. Saying that "There were no WMDs in Iraq!" is isomorphic with saying "There are no white cats on the island".
About these cats, I can state "All cats on the island are black".
This is a strong statement; it says something precise about all the cats of the island. It is also quite easy to disprove: you just have to find one single cat of any colour except black.
So, I may go for a weaker statement: "The vast majority of cats are black".
This is harder to disprove: first, you and I have to agree on a common definition of "vast majority": it can be 2/3, 3/4 etc. Second, you have to count the cats, or do a statistical study in order to back up your assertion (that would be, black cats do not constitute a vast majority).
I may do a statistical study and say: "At 95% confidence, (90 ± 7)% of cats are black".
You do your study too, and conclude that at 95% confidence only (80 ± 5)% of the cats are black. The confidence intervals overlap (even so slightly), so it's quite hard to say which statistics is right. Not to mention that we could accuse each other of having used a flawed method or an unrepresentative sample of the population.
I may also say "There are no white cats on the island".
This is another strong statement, but it's also negative, and this makes things much more complicated. For example, the only white cat may be hiding while you scout the island back and forth searching for it. So a doubt will always remain: you did not see any white cat because there is none, or because you were unable to spot the few present?
Generally, a strong negative statement is rather dishonest intellectually, because of this unresoluble doubt. Not always, sometimes a negative statement is OK; it depends from the context. For example, "There are no electrons with positive charge" is a honest statement, because those particles with positive charge and mass equal to the electron's mass are called positrons.
And all the above supposing that we both are being honest - or at least honest enough.
We may both play dirty, and things will get much more complicated.
For example, when you produce one grey cat I may insinuate that you did not find it on the island, but brought it from the mainland.
I can refuse to agree on a common definition of "vast majority" and thus make discussions about it meaningless.
I may manipulate my statistics, and/or accuse you to do so - this is different from the case discussed above, because the idea of manipulation implies a malicious will; not good-faith mistakes.
Or, when trying to prove the existence of at least one white cat you may publish a dark, out-of-focus, low- resolution photograpy showing a white indistinct blur, and insist that it's the infamous white cat. But it is impossible to tell exactly what the white blur is - accepting that the picture was indeed taken on the island. But when asked to produce better evidence, you say that it's not possible, the white cat won't show itself again but anyway you believe it exists and I should trust you. But my mate Chris says that he lost his white rabbit on the island, just a few hours before the photo was taken...
Pressed further, you then state that the white cat was kidnapped by the Evil Secret Service after you took the picture, because the son of Big Oil Corporation's CEO likes to have his summer holiday on the island, but he loathes white cats and so the CEO used his political connections to have the few remaining white cats kidnapped and killed by the ESS. And save the white cats from the capitalist oppression, too. (It does not make sense, but it isn't very different from popular conspiracy theories).
This is just a fanciful and at times intentionally silly example, but it is based on the rules of logic and, peripherally, set theory. And if you look closely, you may find that it bears some relation with real events. Saying that "There were no WMDs in Iraq!" is isomorphic with saying "There are no white cats on the island".
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