Raven paradox

Black crow.

No-black, no-brain.

The raven paradox was proposed by the German philosopher Carl Hempel in the 1940s to illustrate a problem where inductive logic defies intuition. This paradox is also known as the negation paradox or Hempel's paradox.

Paradox

Hempel describes the paradox in terms of the hypotheses:

(1) All the ravens are black.. In the form of involvement, this can be expressed as: If anything is a raven, then it's black.

By contrast, this phrase is the equivalent of:

(2) If something isn't black, then it's not a raven..

In all circumstances where (2) is true, (1) is true, as well as in all circumstances where (2) is false (for example, in a world where something other than black, but if it were a raven), (1) would also be false.

Given a general phrase such as all crows are black, a form of it that is a specific observable instance of the general class would typically be considered evidence of the general phrase. For example,

(3) The raven I have pet is black.

This is evidence that confirms the hypothesis that all crows are black.

The paradox appears when the same process is applied to sentence (2). Looking at a green apple, one can observe:

(4) The green apple isn't black, so it's not a raven..

By the same reasoning, the sentence evidences that (2) if something is not black, then it is not a raven. But since this sentence is logically equivalent to (1) all crows are black, suggests that seeing a green apple provides evidence that crows are black. This conclusion seems paradoxical because it implies that information about crows has been gained by looking at an apple.

Detailed description

When for thousands of years people have observed facts that fit well within the framework of a theory such as the law of gravity, they tend to believe that the theory has a high probability of being true, and confidence in it increases. with each new observation agreeing with it. This type of reasoning can be synthesized in the principle of induction:

• If a particular case is observed X consistent with theory T, then the probability that T That's true.

Hempel gives an example of the principle of induction. He proposes as a theory "all crows are black." If you examine a million crows and find that they are all black, your belief in the "all crows are black" theory will grow slightly with each observation. In this case, the principle of induction seems reasonable.

Now, the statement "all ravens are black" is logically equivalent to the statement "all non-black things are non-ravens".^{[citation needed]< /sup> Thus, looking at a red apple provides empirical evidence to support this second claim. A red apple is a non-black thing, and when examined, it is seen to be a non-crow. So, by the induction principle, looking at a red apple should increase confidence in the belief that all crows are black.}

There are philosophers who have offered various solutions to this challenge to intuition. The American logician Nelson Goodman has suggested adding restrictions to the reasoning itself, such as never considering that a valid case "all P are Q" also validates "no P is Q".

Other philosophers have questioned the "equivalence principle." Perhaps the red apple should increase belief in the "all non-black things are non-ravens" theory without increasing belief in the "all ravens are black" theory. This suggestion has also been challenged, however, on the grounds that you cannot have different levels of belief in two statements if you know that both are either true or false at the same time. Goodman, and later Quine, used the term projectable predicate to describe expressions, such as raven and black, that yes allow the use of inductive generalizations. The non-projectable predicates are those like no-black and no-crow, which apparently do not allow it (See also verjo, another non-projectable predicate invented by Goodman). Quine suggested that it is an empirical question which, if any, of the predicates are projectable, and notes that in a universe of infinitely many objects, the complement of a projectable predicate must always be non-projectable. This would have the consequence that, although "all ravens are black" and "all non-black things are non-ravens" must be validated at the same time, they both derive their support from black ravens, not from non-ravens. -non-black crows.

Some philosophers have argued that intuition is at fault. Looking at a red apple actually increases the probability that all crows are black. After all, if someone were to show all the non-black things in the universe, and it could be seen that there are no ravens among them, then one could conclude that all ravens are black. The example just defies intuition because the set of non-black things is by far larger than the set of crows. Thus, observing something non-black that is not a crow should change belief in the theory very little when compared to observing another crow that is black.

There is an alternative to the "principle of induction" described above.

Let X be an instance of the theory T, and I all the information about the environment.

Sea ${displaystyle Pr(bullet Δcirc)}$ the probability of ${displaystyle bullet }$ given ${displaystyle circ }$. So,

${displaystyle Pr(TINDXI)={frac {Pr(TINDI)cdot Pr(XATATI)}{Pr(XATAI)}}}}}}$

This principle is known as Bayes' theorem. It is one of the bases of probability and statistics. When scientists publish analyzes of experimental results and find them to be statistically significant or statistically insignificant, they are using this principle implicitly, so it could be argued that this principle better describes scientific reasoning than the original "principle of induction."

If this principle is used, the paradox does not appear. If someone is asked to pick an apple at random and show it, then the probability of seeing a red apple is independent of the color of the crows. The numerator will be equal to the denominator, so the division will be equal to one, and the probability will remain unchanged. Seeing a red apple will not affect the belief that all crows are black.

If someone is asked to pick a non-black thing at random, and they show a red apple, then the numerator will be slightly greater than the denominator. Seeing the red apple will only slightly increase the belief that all crows are black. One would have to see almost everything in the universe (and prove that it is non-raven) for the belief in "all ravens are black" to increase appreciably. In both cases, the result is according to intuition.

Sherlock Holmes and this paradox

"When all impossible explanations have been ruled out, the one that remains, however implausible it may seem, must be the true one", says Sherlock Holmes. At first sight it seems a reasonable statement, since it ultimately refers to to the old and effective method of reduction to absurdity. But there is a problem: Holmes's method presupposes knowing all the competing possibilities in a case, and then ruling out all but one on the basis of its infeasibility, and this amounts to full—that is, divine—knowledge of the situation and its circumstances..

Holmes's fallacy and Hempel's paradox are, to a large extent, due to the fact that they refer to incomprehensible, practically infinite sets, be they the possible explanations of a crime or the non-black objects of the universe.