This problem was proposed by the logician Carl Gustav Hempel in the 1940s to illustrate a contradiction between inductive logic and intuition.
Solutions that accept the paradoxical conclusion can do this by presenting a proposition that we intuitively know to be false but that is easily confused with (PC), while solutions that reject (EC) or (NC) should present a proposition that we intuitively know to be true but that is easily confused with (EC) or (NC).
Although this conclusion of the paradox seems counter-intuitive, some approaches accept that observations of (coloured) non-ravens can in fact constitute valid evidence in support for hypotheses about (the universal blackness of) ravens.
The reason becomes clear when we compare the previous situation with the case of an experiment where an object whose chemical constitution is as yet unknown to us is held into a flame and fails to turn it yellow, and where subsequent analysis reveals it to contain no sodium salt.
In the seemingly paradoxical cases of confirmation, we are often not actually judging the relation of the given evidence, E alone to the hypothesis H ... we tacitly introduce a comparison of H with a body of evidence which consists of E in conjunction with an additional amount of information which we happen to have at our disposal; in our illustration, this information includes the knowledge (1) that the substance used in the experiment is ice, and (2) that ice contains no sodium salt.
But if we are careful to avoid this tacit reference to additional knowledge ... the paradoxes vanish.One of the most popular proposed resolutions is to accept the conclusion that the observation of a green apple provides evidence that all ravens are black but to argue that the amount of confirmation provided is very small, due to the large discrepancy between the number of ravens and the number of non-black objects.
According to this resolution, the conclusion appears paradoxical because we intuitively estimate the amount of evidence provided by the observation of a green apple to be zero, when it is in fact non-zero but extremely small.
Noteworthy approaches using Bayesian techniques (some of which accept !PC and instead reject NC) include Earman,[15] Eells,[16] Gibson,[17] Hosiasson-Lindenbaum,[12] Howson and Urbach,[18] Mackie,[19] and Hintikka,[20] who claims that his approach is "more Bayesian than the so-called 'Bayesian solution' of the same paradox".
Bayesian approaches that make use of Carnap's theory of inductive inference include Humburg,[21] Maher,[8] and Fitelson & Hawthorne.
Maher[8] accepts the paradoxical conclusion, and refines it: A non-raven (of whatever color) confirms that all ravens are black because 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.
Using this Carnapian approach, Maher identifies a proposition we intuitively (and correctly) know is false, but easily confuse with the paradoxical conclusion.
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.
The reason is that the background knowledge that Good and others use can not be expressed in the form of a sample proposition – in particular, variants of the standard Bayesian approach often suppose (as Good did in the argument quoted above) that the total numbers of ravens, non-black objects and/or the total number of objects, are known quantities.
But ... given any sample proposition as background evidence, a non-black non-raven confirms A just as strongly as a black raven does ...
Hempel rejected this as a solution to the paradox, insisting that the proposition 'c is a raven and is black' must be considered "by itself and without reference to any other information", and pointing out that it "was emphasized in section 5.2(b) of my article in Mind ... that the very appearance of paradoxicality in cases like that of the white shoe results in part from a failure to observe this maxim.
Good had shown that, for some configurations of background knowledge, Nicod's criterion is false (provided that we are willing to equate "inductively support" with "increase the probability of" – see below).
The possibility remained that, with respect to our actual configuration of knowledge, which is very different from Good's example, Nicod's criterion might still be true and so we could still reach the paradoxical conclusion.
Hempel, on the other hand, insists our background knowledge itself is the red herring, and that we should consider induction with respect to a condition of perfect ignorance.
Good had used this fact before to respond to Hempel's insistence that Nicod's criterion was to be understood to hold in the absence of background information:[25] ... imagine an infinitely intelligent newborn baby having built-in neural circuits enabling him to deal with formal logic, English syntax, and subjective probability.
Maher made Good's argument more precise by using Carnap's theory of induction to formalize the notion that if there is one raven, then it is likely that there are many.
Quine[27] argued that the solution to the paradox lies in the recognition that certain predicates, which he called natural kinds, have a distinguished status with respect to induction.
This suggests a resolution to the paradox – Nicod's criterion is true for natural kinds, such as "blue" and "black", but is false for artificially contrived predicates, such as "grue" or "non-raven".
[20] Hintikka was motivated to find a Bayesian approach to the paradox that did not make use of knowledge about the relative frequencies of ravens and black things.
Despite the fact that we lack background knowledge to indicate that there are dramatically fewer men than short people, we still find ourselves inclined to reject the conclusion.
Hintikka's example is as follows: "a generalization like 'no material bodies are infinitely divisible' seems to be completely unaffected by questions concerning immaterial entities, independently of what one thinks of the relative frequencies of material and immaterial entities in one's universe of discourse.
Scheffler and Goodman[28] took an approach to the paradox that incorporates Karl Popper's view that scientific hypotheses are never really confirmed, only falsified.
A possible loophole is to interpret "All" as "Nearly all" – "Nearly all ravens are black" is not equivalent to "Nearly all non-black things are non-ravens", and these propositions can have very different probabilities.
As Neyman and Pearson put it: Without hoping to know whether each separate hypothesis is true or false, we may search for rules to govern our behaviour with regard to them, in following which we insure that, in the long run of experience, we shall not be too often wrong.
[34]Several commentators have observed that the propositions "All ravens are black" and "All non-black things are non-ravens" suggest different procedures for testing the hypotheses.
[35] The argument considers situations in which the total numbers or prevalences of ravens and black objects are unknown, but estimated.