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Hempel describes the paradox in terms of a hypothesis that all ravens are black. This statement is equivalent, in logical terms, to the statement that all non-black things are non-ravens. If one were to observe many ravens and find that they were all black, one's belief in the statement that all ravens are black would increase. But if one were to observe many red apples, and concur that all non-black things are non-ravens, one would still become more sure that all ravens are black.
The principle of empirical evidenceEdit
The principle of empiricism states that:
- As an instance X is observed that is consistent with theory T, then the probability that T is true increases
In the Raven paradox, the 'law' being tested is that all ravens are black. This problem has been summarized (derived from a poem by Gelett Burgess) as:
- I never saw a purple cow
- But if I were to see one
- Would the probability ravens are black
- Have a better chance to be one?
The origin of the paradox lies in the fact that the statements "all Ravens are black" and "all non-black things are non-ravens" are indeed equivalent, while the act of finding a black raven is not at all equivalent to finding a non-black non-raven. Confusion is common when these two notions are thought to be identical.
Philosophers have offered many solutions to this violation of intuition. For instance, the American logician Nelson Goodman suggested adding restrictions to our reasoning, such as never considering an instance as support for "All P are Q" unless it also falsifies its logical contrary "No P are Q".
Other philosophers have questioned the "principle of equivalence" between the two theorems. Perhaps the red apple should increase our belief in the theory all non-black things are non-ravens, without increasing our belief that all ravens are black. But in classical logic one cannot have a different degree of belief in two equivalent statements, if one knows that they are either both true or both false.
Goodman, and later another philosopher, Quine, used the term projectible predicate to describe those expressions, such as raven and black, which do allow inductive generalization; non-projectible predicates are by contrast those such as non-black and non-raven which apparently do not. Quine suggests that it is an empirical question which, if any, predicates are projectible; and notes that in an infinite domain of objects the complement of a projectible predicate ought always be non-projectible. This would have the consequence that, although "All ravens are black" and "All non-black things are non-ravens" must be equally supported, they both derive all their support from black ravens and not from non-black non-ravens.
I proved the foundation of it to be incorrect. Statement " All ravens are black" is not logically equivalent to the statement " Anything which is not black is not a raven ".
They actually are equivallent:
A: x is a raven
B: x is black
A -> B ("If x is a raven then x is black", or "All ravens are black") is equivalent to ~A or B, which is B or ~A, which can be converted to a if then statement again: ~B->~A, that is "if x is not black, then x is not a raven" or "all non-blacks are non-ravens" or "anything not black is not a raven".
- Franceschi, P. The Doomsday Argument and Hempel's Problem, English translation of a paper initially published in French in the Canadian Journal of Philosophy 29, 139-156, 1999, under the title Comment l'Urne de Carter et Leslie se Déverse dans celle de Hempel
- Hempel, C. G. A Purely Syntactical Definition of Confirmation. J. Symb. Logic 8, 122-143, 1943.
- Hempel, C. G. Studies in Logic and Confirmation. Mind 54, 1-26, 1945.
- Hempel, C. G. Studies in Logic and Confirmation. II. Mind 54, 97-121, 1945.
- Hempel, C. G. Studies in the Logic of Confirmation. In Marguerite H. Foster and Michael L. Martin, eds. Probability, Confirmation, and Simplicity. New York: Odyssey Press, 1966. 145-183.
- Whiteley, C. H. Hempel's Paradoxes of Confirmation. Mind 55, 156-158, 1945.