The Paradox of Hempel was proposed by the German logician Carl Gustav Hempel in the years 1940 to illustrate the fact that the inductive logical could violate the intuition. This paradox is also named paradox of the corbel or ornithology in room .

Statement

If I say “All the corbels are black”, this sentence is logically equivalent to “All the not-blacks objects are not-corbels” (law of contraposition: P=>Q is equivalent to non-Q => non-P). To reinforce by a process of induction my conviction that “All the not-blacks objects are not-corbels”, I can extremely well remain in my room, to find ten thousand objects nonblack, and check there that they all are well of the not-corbels. Isn't a law which is checked on ten thousand observations without the least exception is certainly valid?

Solutions suggested

Solution bayesienne

To start, and to simplify, we will suppose in this article that all the corbels are black, without exception.

In fact, it is to better leave its room to show that all the corbels are black. To show it, it is necessary to use the Théorème of Bayes

Let us call $T$ the theory, and $X$ a practical example confirming the theory. The question is to know if $X$ makes it possible to make $T$ more probable.

• In the system with the great outdoors , $T$ is the proposal: all the corbels are black.

And $X$ the proposal: I found a corbel, and it is black.

• In the system in room , $T$ is the proposal: All the not-blacks are not-corbels.

And $X$ the proposal. I found a not-black object, and not-corbel (for example a house plant).

In these two systems, by applying the theorem of Bayes:

$P\; (T|X)\; =\; P\; (X|T)\; \backslash \; frac\; \{P\; (T)\}\{P\; (X)\}$

$P\; (X|T)\; =\; \backslash ,\; 1$

because if the theory is true, then a given particular case checks certainly the theory.

Thus:

$P\; (T|X)\; =\; P\; (T)\; \backslash \; frac\; \{1\}\; \{P\; (X)\}$

In the system " with large the air" , $P\; (X)$ is the prior probability that a corbel is black. It is weak, because many other colors are possible.

In the system " in chambre" , $P\; (X)$ is the probability that a not-black object is not-corbel. It is almost a certainty, because the majority of the objects of the world are not corbels.

Consequently, in the system with the great outdoors,

$P\; (T|X)\; >\; \backslash ,\; P\; (T)$

Whereas in the system in room,

$P\; (T|X)\; \backslash \; approx\; P\; (T)$

The only way of returning the theory that all the corbels are black more probable is to leave. To remain at home and to look at its house plant make it possible, curiously, to make the theory more probable, but the progression is negligible. contained in my room are not-corbels. In logical inductive, it is always necessary to specify the context of an observation (see Inférence bayésienne and Conditional probability). -->

See too

Related articles

• Paradox of Goodman
• Paradox
• List of paradoxes
• Theorem of Cox-Jaynes

External bonds

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References

• Nicholas Falletta, the book of the paradoxes , ED. Diderot, 1998. ISBN.
• Paul Franceschi, '' How the ballot box of Casing and Leslie flows in that of Hempel '', Canadian Journal off Philosophy, Vol.29, 1999, pp. 139-156

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