The solution proposed by Hempel to the Ravens' Paradox

The evidence expressed by the phrase "This hat is green" confirms the hypothesis that “all ravens are black”?
　　 Let's clarify how the counter-intuitive result of the evidence expressed by the phrase "This hat is green" confirms the hypothesis that "All ravens are black" derived from the solution proposed by Hempel to the Ravens' Paradox.
　　 We will conclude by saying that Hempel's response to this counter-intuitive result owes much to his methodological assumption, however, to exclude from the confirmation relationship any and all additional information about the world - any and all background information.
　　 However, we may disagree with this assumption as being epistemologically unrealistic, since it is on the additional information about the world that we actually make generalizations and predictions.
　　 The objective of the so called Confirmation’s Theory - briefly characterized - is to formalize how the evidence expressed by phrases such as "This raven is black" confirmed or unconfirmed general hypothesis of empirical nature such as "All ravens are black", increasing (if confirmed) or decreasing (if unconfirmed) our reliability in it or leaving us at all indifferent to it (in which case they say, of the evidence expressed in this manner, that the evidence is neutral to this hypothesis).
　　 We live relying on induction.
　　 For example, from ‘Each of the observed F so far is G’ we generalize ‘All F are G’ or anticipate ‘The next F you observe will also be G’.
　　 The truth of the premises of an inductive argument never excluded the logical possibility of the conclusion to be false.
　　 She (induction) is inherently unreliable. However, the assumption shared by the theorists of confirmation is that, if done carefully, inductive reasoning is sufficiently reliable to justify the belief in empirical generalizations or predictions.
　　 The confirmation relationship is conceived by them (theorists of confirmation) as counterpart of the relation of logical consequence typical of deductive reasoning in which necessarily, if the premises are true, the conclusion is true.
　　 The following three conditions seem to be basics principles of Confirmation’s Theory:

         A. a hypothesis H, ‘All F are G’, it is confirmed by the evidence E expressed by singular propositions of the form ‘This is a F and is a G’, otherwise, if the evidence E expressed by singular propositions is of the form ‘This is a F and is a non-G’, the evidence E thus expressed unconfirmed that hypothesis H;

        B. the evidence E expressed by singular propositions of the form ‘This is a non-F and is a G’ or ‘This is a non-F and is a non-G’ neither confirms nor disconfirms H;

        C. if E confirms H, then E confirm any hypothesis logically equivalent to H and if H is confirmed by E, so H is confirmed by any proposition logically equivalent to E.

    These three conditions, respectively known in the literature by Nicod condition (A), No Confirmation condition (B) and Equivalence condition (C), seem extremely plausible and even elementary.
    However, from the conjunction of those three conditions A, B, C it is possible to derive a contradiction.
    Let's see then how a contradiction can be derived from the conjunction of those three conditions A, B, C.
    Consider the condition A (Nicod). For example, the evidence expressed by the phrase "This thing is non-black and is non-raven", confirms the general hypothesis of empirical nature "All things non-black are non-ravens", that is, we need to go to the world, to experience and, according to Nicod’s condition, every time we got something non-black and non-raven we're confirming the generalization "All things non-black are non-ravens".
    Now, by an elementary logical result, the general hypothesis of empirical nature "All things non-black are non-ravens” is equivalent to the general hypothesis of empirical nature "All ravens are black".

From here, from these two steps (from Nicod's condition and from the logical result of the previous paragraph) and from condition C (from Equivalence) it follows that the evidence expressed by the phrase "This thing is non-black and is non-raven" confirms the general empirical hypothesis "All ravens are black".
　 Now, by B’s condition (No Confirmation), the evidence expressed by the phrase "This thing is non-raven and is non-black" does not confirm the general hypothesis of empirical nature "All ravens are black".
However, according to another elementary logical rule, the evidence expressed by the phrase "This thing is non-black and is non-raven" is equivalent to the evidence expressed by the phrase "This thing is non-raven and is non-black".
From here and the condition C (Equivalence) it follows that the evidence expressed by the phrase "This thing is non-raven and is non-black" confirms the general hypothesis of empirical nature "All ravens are black".
So, from the conjunction of those three conditions A, B, C, we derive a contradiction.
　 Nicod’s condition (A), No Confirmation condition (B) and the Equivalence condition (C), even though each one of them individually seems extremely plausible and elementary, their conjunction is not at all sheltered from any objection.
　 In particular, as we just have shown, you can derive from their conjunction the following contradiction: at the same time the evidence expressed by the phrase "This thing is non-raven and is non-black" confirms and does not confirm the general hypothesis of empirical nature "All ravens are black".
Any of those three conditions A, B, C should not be fine. If, from their conjunctions, you can derive a contradiction, some of them must be wrong.
This result, discovered by the philosopher of science Carl G. Hempel (1905-1997), is known in Confirmation’s Theory as the Ravens' Paradox.
　 We have implicitly defined what is a paradox, explicitly we can say that, in the strict sense, a paradox is a seemingly valid argument (or strong) with seemingly true premises and a contradictory conclusion.
　　 Let's see now how Hempel proposes that we solved the Ravens' Paradox.
　　 The logical equivalences between "All things non-black are non-ravens" and "All ravens are black" and between "This thing is non-black and is non-raven" and "This thing is non-raven and is non-black" cannot be seriously disputed.
　　 The condition C (Equivalence) also cannot be seriously disputed.
　　 If any of those three conditions A, B, C must be wrong, if we exclude the C, the mutually incompatible conditions can only be the A (Nicod) and B (Not Confirmation).
　　 It seems that, to solved the paradox, we have only the following two paths: either dropped the Nicod’s condition (A) or dropping the Not Confirmation’s condition (B).
　　 The solution to the Ravens' Paradox proposed by Hempel himself was to drop the No Confirmation’s condition (B), keeping as compatible principles of Confirmation’s Theory, the Nicod’s condition (A) and the Equivalency’s condition (C).
　　 And in fact it is so, from the conjunction of A and C it is not possible to derive a contradiction.
　　 The contradiction we derive was that "This thing is non-raven and is non-black" simultaneously confirms and does not confirm "All ravens are black".
　　 However, if (as Hempel proposed) we dropped the No Confirmation’s condition (B), we dropped its result of "This thing is non-raven and is non-black" can't confirm "All ravens are black" and so we eliminate the contradiction.
　　 However, let's see a little more in detail Hempel’s proposal in maintaining as compatible principles of Confirmation’s Theory the Nicod’s condition (A) and the Equivalency’s condition (C).
　　 According to both, "This thing is non-black and is non-raven" confirms "All things non-black are non-ravens" (condition A) and "This thing is non-raven and is non-black" confirms "All ravens are black" (condition C).
　　 Apparently everything is fine. "This thing is non-black and is non-raven" is logically equivalent to "This thing is non-raven and is non-black" and "All things non-black are non-ravens” is logically equivalent to "All ravens are black".
　　 However, now it seems that we have the following counter-intuitive result.
　　 Consider, for example, "This thing is non-raven and is non-black" confirms "All ravens are black" (condition C) - but given the mentioned logical equivalences, for the condition A is the same thing.
　　 What the condition (C) authorizes us to say, to use the example of the first paragraph of our essay, is that the evidence expressed by the phrase "This hat is green" confirms the hypothesis that “All ravens are black”.
　　 Hempel’s solution to the Ravens' Paradox eliminates the contradiction in the way we have just seen, the paradox (strict sense) disappears but the paradox does not disappear in the broadest sense of having the counter-intuitive result of the evidence expressed by the phrase "This hat is green" confirms the hypothesis that all ravens are black.
　　 However, for Hempel, the evidence expressed by the phrase "This hat is green" confirms the hypothesis that “all ravens are black” is not a counter-intuitive result and its counterintuitiveness derives from a psychological illusion whose nature is captured in the following way.
　　 Hempel's idea is that this result seems counter-intuitive because we consider the evidence expressed by singular propositions with the form ‘This is a non-F and is a non-G’ in the much wider scope of the additional information about the world that each of us has available.
　　 However, for Hempel, this is a psychological illusion.
　　 The confirmation relationship of the Theory that formalises the way the evidence expressed by phrases such as "This thing is non-raven and is non-black" confirms general hypotheses of empirical nature such as "All ravens are black" is not to be conceived, argues Hempel, with any and all the additional information associated psychologically (so to speak) for us to the confirmation relationship, but, by analogy to the relation of logical consequence, is to be conceived independently and autonomously from any and all additional information that psychologically we associate with the confirmation relationship, by chance.
　　 In other words, in the same way that the consideration of additional information about the world is at all irrelevant to the relation of logical consequence, is also irrelevant to the confirmation relationship.
　　 For Hempel, the principles of the Confirmation Theory to be preserved (because from its conjunctions will not be possible to derive a contradiction) are the Nicod’s condition (A) and the Equivalency’s condition (C).
　　 The condition C, for Hempel, just seems to generate a counter-intuitive result. But this result is illusory.
　　 According to the condition A, the observation of things that are non-black and non-ravens confirms the generalization "All non-black things are non-ravens". This generalization is logically equivalent to "All ravens are black".
　　 Now, by C’s condition, the observation of things that are non-black and non-ravens will also have to confirm the generalization "All ravens are black" (if E confirms H, then E confirm any hypothesis logically equivalent to H and if H is confirmed by E, so H is confirmed by any proposition logically equivalent to E).
　　 Here, for Hempel, we are not including any and all additional information about the world and, in this sense, there is nothing here in what our intuition can be based to controvert this Confirmation’s Theory result, at least nothing we can oppose to the logic of reasoning there involved.
　　 And let's conclude.
　　 Hempel’s reply to result of counterintuitiveness owes however much to his methodological assumption (described above) of excluding from the confirmation relationship any and all additional information about the world - any and all background information.
　　 However, we can disagree with this assumption as being epistemologically unrealistic (such as defends, for example, by Mackie, J. L.,1963, "The Paradox of Confirmation", The British Journal for the Philosophy of Science, XIII, 52, 265-77).
　　 Briefly, it's on this background that at any time we make the generalizations and predictions that in fact we make.
　　 So is much more realistic, in epistemic terms, an assumption such as the Bayesian, we can even concede to Hempel that the evidence expressed by the phrase "This hat is green" confirms the hypothesis that “All ravens are black”, but given the additional information about the world, say from this confirmation that it is at all insignificant, we can even ignore it totally, since compared to "This hat is green", "This raven is black" confirms "All ravens are black" in a much higher degree.

        References

        Hempel, C. G. (1937), “Le Probleme de la Verite”, Theoria, 3, 206-46.
        Hempel, C.G. (1943), “A Purely Syntactical Definition of Confirmation”, Journal of Symbolic Logic, 8,  122-43.
        Hempel, C. G. (1945), "Studies in the Logic of Confirmation", Mind, 54, 1-26.
        Mackie, J. L. (1963), "The Paradox of Confirmation", The British Journal for the Philosophy of Science, XIII, 52, 265-77.