A paradox is a parody on proof. It begins with realistic premises, but the conclusion falsifies these premises. Therefore, you cannot take a consistent position towards a paradox.

One of the best known paradoxes is the liar-paradox, which comes down to the statement "This sentence is false". Other types are

the Russell paradox, which has two main variants: the barber and the library story

the twin paradox, resulting from Einstein's Relativity Theory

In this short presentation, two main paradox types will be discussed:

those who have to do with induction: Hempel's paradox and the 99 feet man paradox

Of course, extensive treatment of these topics would lead too far. Therefore, a short reference list is provided.

Induction and its family

Hempel's Raven

Open here I flung the shutter, when, with many a flirt and flutter,

In there stepped a stately raven of the saintly days of yore;

Not the least obeisance made he; not a minute stopped or stayed he;

But, with mien of lord or lady, perched above my chamber door

Perched upon a bust of Pallas just above my chamber door

Perched, and sat, and nothing more.

(E. A. Poe, 1845, The Raven)

Are all ravens black? Even the raven in Poes The Raven? The only way to know this is to use induction. Or isn't it?

What if we reason like on this figure, known as Hempel's paradox?

And, what's even more, the red apple also supports the statement that all ravens are white. Clearly, our intuition about induction is false . . . and that's what Nicod led to formulate his criterion for induction.

Or, we can try to avoid the paradox by showing asymmetries between the statements "all ravens are black" and "all non-black things are non-ravens". That's what's tried by incremental confirmation theory.

Still another approach, Carnap's Requirement of complete evidence, is derived from the so-called paradox of the 99 feet man.

Incremental confirmation

The incremental confirmation theory is sometimes called for when Hempel's paradox pops up.

The basic idea is to use probability estimated with Pascal's Rule as the basis for induction (Reichenbach's "straight rule" for estimation). But of course, even a single white raven falsifies the hypothesis that all ravens are black.

Suppose that the total number of ravens in the whole world is about 500,000. Therefore, following Reichenbach, the observation of one black raven would turn the probability of the hypothesis from 0 to 1/500,000. That is at least something. But the observation of a non-black non-raven would increment the probability with 1/N, where N is the total number of objects in the world ( @ 10 80 , if we count every atom apart). Too small a figure to count in.

Of course, this probability game is nonsense, for even a small paradox remains a paradox. Other solutions which were proposed were the Nicod Criterion and the requirement of complete evidence, based upon another paradox: the 99 feet man.

Nicod's criterion

How to avoid the Hempel paradox?

Nicod's criterion for induction simply states that

1. Black ravens make the generalisation "all ravens are black" more probable

2. Non-black ravens falsify the generalisation

3. Non-black non-ravens are irrelevant

With every scientific research, Nicod's criterion comes into play. If that is not the case, we' re in serious troubles.

But, as shown by W.C. Salmon, even this criterion may fail. Another approach is the requirement of Complete Evidence, proposed by Rudolf Carnap. Still another way is formulated by Reichenbach, with his incremental confirmation theory.

The 99 feet man

A beautiful induction paradox is Paul Berents story about the statement "All people are shorter than hundred feet". Suppose you agree with this, until you encounter a man of 99 feet. Your belief in the original statement is somewhat weakened, not? The strange thing is, the 99 feet man also supports the 100 feet hypothesis.

Two causes for this paradox exist. In the first place, we do not always say what we mean. If you use SI measures, you might have said "All people are shorter than 30 metres". Since 98.43 feet is 30 metres, your observation falsifies the hypothesis.

The second cause is that the fact that 99 feet people exist means that very exceptional things might happen in people's growth. And, if today 99 feet, why not tomorrow 100 feet? All human features tend to reappear scaled up, so indeed 100 feet is possible. This can be seen as an application of Carnap's requirement of complete evidence.

Requirement of complete evidence

If you want to reason in an inductive way, you must use all possible evidence. In the 99 feet paradox, simple application of the fact that 100 is larger than 99 is not enough, you have to use your knowledge that 99 feet also means exceptional human growth, thus that the hypothesis of 100 feet is falsified. In the Hempel paradox, the same principle applies. We do not simply divide the world into black and non-black, or raven and not-raven. We know a lot more. If we meet a white crow, our original belief in the black raven hypothesis will be weakened, because we know crows and ravens share a lot of properties.

The requirement of complete evidence was proposed by Rudolf Carnap, and it applies to many different scientific domains such as biochemistry, astronomy and physics. In those disciplines, it is usual to meet phenomena that are almost invisible. Nobody has ever seen an atom in a direct way, or a black hole. This is not wrong, but Carnap's principle states that it might not be a good idea to limit our possibilities of knowledge acquisition. Therefore, independent evidence becomes important.

Categories

It is not so easy to state what categories are. Category members tend to be quite alike each other, while they mostly do not resemble very well other categories' members.

Man is an animal that invents categories, and therefore even the distinction between "natural categories" and "artificial categories" is fuzzy. For most people, division of the category "animal" into (elephants, horses, worms, beetles, sheep, . . .) will surely sound less strange than biologist's division into 22 categories, consisting mostly of worms, and all "normal" animals (sheep, dogs, humans, chickens) belonging to one small substem of one category.

Maybe, natural and artificial language do not divide sharply. Maybe, science's impact on natural language is greater than sometimes thought. Or smaller, for people tend to have difficulties with classifying penguins with birds and whale with mammals.

In any case, our intuition about categories is challenged by a beautiful paradox, invented in 1953 by Nelson Goodman: the famous grue-bleen paradox.

Artificial language

Borges describes an analytic language, where for example "salmon" is translated to "zana". For those who know the category system used, zana in fact means "red freshwater fish with scales". That tells a lot more than "salmon", a natural language word that exists due to historic development, but has no explicit relationship with the thing it refers to.

An analytic language such as described by Borges surely sound crazy. But a strange paradox, first described by Nelson Goodman, in fact uses artificial categories. Since it was stated in 1953, the grue-bleen paradox has been discussed over and over again. Even today, it serves as a benchmark for philosophical theories of induction.

The artificial language described by Borges may seem a little bit far-fetched, but in fact, scientific jargon often is no less weird: to give only one example, classification of quarks uses terminology like "charm" or "strange" quarks. Will these concepts ever be part of laymen's speech, as is for example the case with "gravity" or "the unconscious"?

Grue-Bleen

A jeweller looks at an emerald, and concludes it is green. Aha, he says, I have been working with emeralds for thirty years, and all of them were green. Therefore, I say that every emerald is green.

His collegue, who does not speak English, but Gruebleen, a strange language, also looks at an emerald. His conclusion is different. Since in his language, "grue" means "green before this century changes, and blue thereafter", he would conclude that all emeralds are grue. The paradox is that both jewellers have the same experience, and use the same inductive reasoning, but come to different conclusions: with the century change (December, 31 in 2000, sic), one of the conclusions will be falsified.

At first sight, "grue" is a strange word. Why would an emerald change colour? That does not seem natural, so the Gruebleen-speaker is wrong. But he can say the same thing about our language: for him, green means "grue before the century change and bleen afterwards". Other evident solutions of the paradox fail for the same reason.

To get rid of this, we have to find a non-symmetric case. A case where English and Gruebleen cannot be translated into each other. Such case is made by Goodman himself, and deals with what he calls "projectibility", and involves an argument about the entrenchment of our concepts in natural language.

Projectibility

The only valid objection to the grue-bleen paradox is made by the inventor himself. Goodman says that a concept like "grue" is not projectible, because it does not refer to the real world.

Although certain languages, like the indian Choctaw language, do not have a different word for blue and green. But no natural language concept means "grue" or something like that. Even concepts like "regulatic", proposed by Ackeroid, do not really get the point because the change from regular to erratic behaviour does not happen at the same time for all systems.

A problematic feature such as "grue" is not projectible because it cannot be used in an inductive argument. Science must be aware of non-projectible concepts. For example, most theories about quantum mechanics postulate that quarks exist, but that it is impossible to observe them apart.

Note: because of the large debate concerning this paradox, bleen and grue can already be found in some dictionaries. This strengthens my point that there is no sharp distinction between natural and artificial language.

Natural language

An interesting question following from the entrenchment concept needed to explain projectibility, concerns the nature of natural language concepts.

A useful approach is formulated by Reeke et al., when they describe a system, composed of neural networks and governed by a Darwinist learning rule, is able to achieve very simple category concepts.

Although their study is seriously limited in several ways, I think it provides evidence for the projectibility explanation of the grue-bleen paradox. An autonomous system, called Darwin III, provided with no more than the built-in "rough, striped objects are not healthy" succeeds in differentiating them from other objects. The only feedback given to the system is a negative mark when it makes an error, and a positive when it behaves correctly.

Interesting is that the categorisation is inherently a behavioural act, and not a pattern matching or list comparison algorithm. This sheds light on the projectibility: no Darwin III variant will come up with concepts like "stripooth" (striped and smooth) and the like. We can safely assume that Darwin will develop concepts more or less the same as ours (or Choctaw, an Indian language which does not make differences between green and blue).

Literature

So many interesting topics were left out here. Almost too large a number to sum up. But the interested reader will forgive me if I include a small bibliography, where he or she surely finds everything one could possibly want to know about paradoxes in induction and categories.

Akeroyd, F. M. (1991). A Practical Example of Grue. Brit. J. Phil. Sci, 42(4), 535-539.

Gonick, L. (1989). Neo-babelonia: A serious study in contemporary confusion. Utrecht: Veen.

Gottlob, R. (1995). Emeralds are no Chameleons: Why "Grue" is not Projectible for Induction. J. Gen. Phil. Sci, 26(2), 259-268.

Poundstone, W. (1990). Het labyrint van het denken: Paradoxen, puzzels en de broosheid van kennis. Amsterdam: Contact.

Putnam, H. (1996). Craigh's Theorem. J. Phil, 62, 251-259.

Reeke, G.N., Finkel, L.H., Sporns, O., & Edelman, G.M. (1990). Synthetic neural modelling: A multilevel approach to the analysis of brain complexity. In G.M. Edelman, W.E. Gall, & W.M. Cowan (Eds.), Signal and sense: Local and global order in perceptual maps. NY: Wiley.

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