Editing Raven paradox (section)

====Carnap approach====
Maher<ref name=Maher1999/> accepts the paradoxical conclusion, and refines it:

{{Quote|A non-raven (of whatever color) confirms that all ravens are black because

{{unbulleted list|item_style=margin-left: 1.5em; text-indent: -1.5em|(i) the information that this object is not a raven removes the possibility that this object is a counterexample to the generalization, and
|(ii) it reduces the probability that unobserved objects are ravens, thereby reducing the probability that they are counterexamples to the generalization.}}}}

To reach (ii), he appeals to Carnap's theory of inductive probability, which is (from the Bayesian point of view) a way of assigning prior probabilities that naturally implements induction. According to Carnap's theory, the posterior probability, <math>P(Fa|E)</math>, that an object, <math>a</math>, will have a predicate, <math>F</math>, after the evidence <math>E</math> has been observed, is:

<math display="block">P(Fa|E) \ = \ \frac{n_F+\lambda P(Fa)}{n+\lambda}</math>

where <math>P(Fa)</math> is the initial probability that <math>a</math> has the predicate <math>F</math>; <math>n</math> is the number of objects that have been examined (according to the available evidence <math>E</math>); <math>n_F</math> is the number of examined objects that turned out to have the predicate <math>F</math>, and <math>\lambda</math> is a constant that measures resistance to generalization.

If <math>\lambda</math> is close to zero, <math>P(Fa|E)</math> will be very close to one after a single observation of an object that turned out to have the predicate <math>F</math>, while if <math>\lambda</math> is much larger than <math>n</math>, <math>P(Fa|E)</math> will be very close to <math>P(Fa)</math> regardless of the fraction of observed objects that had the predicate <math>F</math>.

Using this Carnapian approach, Maher identifies a proposition we intuitively (and correctly) know is false, but easily confuse with the paradoxical conclusion. The proposition in question is that observing non-ravens tells us about the color of ravens. While this is intuitively false and is also false according to Carnap's theory of induction, observing non-ravens (according to that same theory) causes us to reduce our estimate of the total number of ravens, and thereby reduces the estimated number of possible counterexamples to the rule that all ravens are black.

Hence, from the Bayesian-Carnapian point of view, the observation of a non-raven does not tell us anything about the color of ravens, but it tells us about the prevalence of ravens, and supports "All ravens are black" by reducing our estimate of the number of ravens that might not be black.