Raven paradox
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The Raven paradox, also known as Hempel's paradox or Hempel's ravens is a paradox proposed by the German logician Carl Gustav Hempel in the 1940s to illustrate a problem where inductive logic violates intuition. It reveals the problem of induction.
Contents 
[edit] The paradox
Hempel describes the paradox in terms of the hypothesis^{[1]}^{[2]}:
In strict logical terms, via the Law of Implication, this statement is equivalent to:
 (2) Everything that is not black is not a raven.
It should be clear that in all circumstances where (2) is true, (1) is also true; and likewise, in all circumstances where (2) is false (i.e. if we imagine a world in which something that was not black, yet was a raven, existed), (1) is also false. This establishes logical equivalence.
Given a general statement such as all ravens are black, we would generally consider a form of the same statement that refers to a specific observable instance of the general class to constitute evidence for that general statement. For example,
 (3) Nevermore, my pet raven, is black.
is clearly evidence supporting the hypothesis that all ravens are black.
The paradox arises when this same process is applied to statement (2). On sighting a green apple, we can observe:
 (4) This green (and thus not black) thing is an apple (and thus not a raven).
By the same reasoning, this statement is evidence that (2) everything that is not black is not a raven. But since (as above) this statement is logically equivalent to (1) all ravens are black, it follows that the sight of a green apple offers evidence that all ravens are black.
[edit] Proposed Resolutions
Two apparently reasonable premises:
 The Equivalence Condition (EC): If a proposition, X, provides evidence in favor of another proposition Y, then X also provides evidence in favor of any proposition which is logically equivalent to Y.
and
 Nicod's Criterion (NC): A proposition of the form "All P are Q" is supported by the observation of a particular P which is Q.
can be combined to reach the seemingly paradoxical conclusion:
 (PC): The observation of a green apple provides evidence that all ravens are black.
A resolution to the paradox must therefore either accept (PC) or reject (EC) or reject (NC) or reject both. A satisfactory resolution should also explain why there naively appears to be a paradox. Solutions which accept the paradoxical conclusion can do this by presenting a proposition which we intuitively know to be false but which is easily confused with (PC), while solutions which reject (EC) or (NC) should present a proposition which we intuitively know to be true but which is easily confused with (EC) or (NC)^{[3]}.
[edit] Approaches which Accept the Paradoxical Conclusion
[edit] Hempel's Resolution
Hempel himself accepted the paradoxical conclusion, arguing that the reason the result appears paradoxical is because we possess prior information without which the observation of a nonblack nonraven would indeed provide evidence that all ravens are black.
He illustrates this with the example of the generalization "All sodium salts burn yellow", and asks us to consider the observation which occurs when somebody holds a piece of pure ice in a colorless flame which does not turn yellow.
 This result would confirm the assertion, "Whatever does not burn yellow is no sodium salt", and consequently, by virtue of the equivalence condition, it would confirm the original formulation. Why does this impress us as paradoxical? 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. This outcome, we should no doubt agree, is what was to be expected on the basis of the hypothesis ... thus the data here obtained constitute confirming evidence for the hypothesis.
 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. If we assume this additional information as given, then, of course, the outcome of the experiment can add no strength to the hypothesis under consideration. But if we are careful to avoid this tacit reference to additional knowledge ... the paradoxes vanish.^{[1]}
[edit] The Standard Bayesian Solution
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 nonblack 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 nonzero but very small.
I J Good's presentation of this argument in 1960^{[4]} is perhaps the best known, and variations of the argument have been popular ever since ^{[5]} although it had been presented in 1958^{[6]} and early forms of the argument appeared as early as 1940^{[7]}.
Good's argument involves calculating the weight of evidence provided by the observation of a black raven or a white shoe in favor of the hypothesis that all the ravens in a collection of objects are black. The weight of evidence is the logarithm of the Bayes factor, which in this case is simply the factor by which the odds of the hypothesis changes when the observation is made. The argument goes as follows:
... suppose that there are N objects that might be seen at any moment, of which r are ravens and b are black, and that the N objects each have probability 1/N of being seen. Let H_{i} be the hypothesis that there are i nonblack ravens, and suppose that the hypotheses H_{1},H_{2},...,H_{r} are initially equiprobable. Then, if we happen to see a black raven, the Bayes factor in favour of H_{0} is
 average
i.e. about 2 if the number of ravens in existence is known to be large. But the factor if we see a white shoe is only
 average
 and this exceeds unity by only about r/(2N2b) if Nb is large compared to r. Thus the weight of evidence provided by the sight of a white shoe is positive, but is small if the number of ravens is known to be small compared to the number of nonblack objects.^{[8]}
Many of the proponents of this resolution and variants of it have been advocates of Bayesian probability, and it is now commonly called the Bayesian Solution, although, as Chihara^{[9]} observes, "there is no such thing as the Bayesian solution. There are many different 'solutions' that Bayesians have put forward using Bayesian techniques." Noteworthy approaches using Bayesian techniques include Earman, ^{[10]}, Eells ^{[11]}, Gibson ^{[12]}, HosaissonLindenbaum ^{[13]}, Howson and Urbach ^{[14]}, Mackie ^{[15]} and Hintikka^{[16]}, who claims that his approach is "more Bayesian than the socalled 'Bayesian solution' of the same paradox." Bayesian approaches which make use of Carnap's theory of inductive inference include Humburg^{[17]}, Maher, ^{[18]} and Fitelson et al.^{[19]}. Vranas^{[20]} introduced the term "Standard Bayesian Solution" to avoid confusion.
[edit] The Carnapian Approach
Maher^{[21]} accepts the paradoxical conclusion, and refines it:
A nonraven (of whatever color) confirms that all ravens are black because
 (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.
In order 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 which naturally implements induction. According to Carnap's theory, the posterior probability, P(Fa  E), that an object, a, will have a predicate, F, after the evidence E has been observed, is:
where P(Fa) is the initial probability that a has the predicate F; n is the number of objects which have been examined (according to the available evidence E); n_{F} is the number of examined objects which turned out to have the predicate F, and λ is a constant which measures resistance to generalization.
If λ is close to zero, P(Fa  E) will be very close to one after a single observation of an object which turned out to have the predicate F, while if λ is much larger than n, P(Fa  E) will be very close to P(Fa) regardless of the fraction of observed objects which had the predicate F.
Using this Carnapian approach, Maher identifies a proposition which we intuitively (and correctly) know to be false, but which we easily confuse with the paradoxical conclusion. The proposition in question is the proposition that observing nonravens tells us about the color of ravens. While this is intuitively false and is also false according to Carnap's theory of induction, observing nonravens (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 BayesianCarnapian point of view, the observation of a nonraven 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 which might not be black.
[edit] The Role of Background Knowledge
Much of the discussion of the paradox in general and the Bayesian approach in particular has centred on the relevance of background knowledge. Surprisingly, Maher^{[21]} shows that, for a large class of possible configurations of background knowledge, the observation of a nonblack nonraven provides exactly the same amount of confirmation as the observation of a black raven. The configurations of background knowledge which he considers are those which are provided by a sample proposition, namely a proposition which is a conjunction of atomic propositions, each of which ascribes a single predicate to a single individual, with no two atomic propositions involving the same individual. Thus, a proposition of the form "A is a black raven and B is a white shoe" can be considered a sample proposition by taking "black raven" and "white shoe" to be predicates.
Maher's proof appears to contradict the result of the Bayesian argument, which was that the observation of a nonblack nonraven provides much less evidence than the observation of a black raven. The reason is that the background knowledge which 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, nonblack objects and/or the total number of objects, are known quantities. Maher comments that, "The reason we think there are more nonblack things than ravens is because that has been true of the things we have observed to date. Evidence of this kind can be represented by a sample proposition. But ... given any sample proposition as background evidence, a nonblack nonraven confirms A just as strongly as a black raven does ... Thus my analysis suggests that this response to the paradox [i.e. the Standard Bayesian one] cannot be correct."
Fitelson et al.^{[22]} examined the conditions under which the observation of a nonblack nonraven provides less evidence than the observation of a black raven. They show that, if a is an object selected at random, Ba is the proposition that the object is black, and Ra is the proposition that the object is a raven, then the condition:
is sufficient for the observation of a nonblack nonraven to provide less evidence than the observation of a black raven. Here, a line over a proposition indicates the logical negation of that proposition.
This condition does not tell us how large the difference in the evidence provided is, but a later calculation in the same paper shows that the weight of evidence provided by a black raven exceeds that provided by a nonblack nonraven by about . This is equal to the amount of additional information (in bits, if the base of the logarithm is 2) which is provided when a raven of unknown color is discovered to be black, given the hypothesis that not all ravens are black.
Fitelson et al.^{[22]} explain that:
 Under normal circumstances, may be somewhere around 0.9 or 0.95; so 1 / p is somewhere around 1.11 or 1.05. Thus, it may appear that a single instance of a black raven does not yield much more support than would a nonblack nonraven. However, under plausible conditions it can be shown that a sequence of n instances (i.e. of n black ravens, as compared to n nonblack nonravens) yields a ratio of likelihood ratios on the order of (1 / p)^{n}, which blows up significantly for large n.
The authors point out that their analysis is completely consistent with the supposition that a nonblack nonraven provides an extremely small amount of evidence although they do not attempt to prove it; they merely calculate the difference between the amount of evidence that a black raven provides and the amount of evidence that a nonblack nonraven provides.
[edit] Rejecting Nicod's Criterion
[edit] The Red Herring
Good ^{[23]} gives an example of background knowledge with respect to which the observation of a black raven decreases the probability that all ravens are black:
 Suppose that we know we are in one or other of two worlds, and the hypothesis, H, under consideration is that all the ravens in our world are black. We know in advance that in one world there are a hundred black ravens, no nonblack ravens, and a million other birds; and that in the other world there are a thousand black ravens, one white raven, and a million other birds. A bird is selected equiprobably at random from all the birds in our world. It turns out to be a black raven. This is strong evidence ... that we are in the second world, wherein not all ravens are black.
Good concludes that the white shoe is a "red herring": Sometimes even a black raven can constitute evidence against the hypothesis that all ravens are black, so the fact that the observation of a white shoe can support it is not surprising and not worth attention. Nicod's criterion is false, according to Good, and so the paradoxical conclusion does not follow.
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."^{[24]}
The question which then arises is whether the paradox is to be understood in the context of absolutely no background information (as Hempel suggests), or in the context of the background information which we actually possess regarding ravens and black objects, or with regard to all possible configurations of background information.
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 that it is our background knowledge itself which is the red herring, and that we should consider induction with respect to a condition of perfect ignorance.
[edit] Good's Baby
In his proposed resolution, Maher implicitly made use of the fact that the proposition "All ravens are black" is highly probable when it is highly probable that there are no ravens. 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 builtin neural circuits enabling him to deal with formal logic, English syntax, and subjective probability. He might now argue, after defining a raven in detail, that it is extremely unlikely that there are any ravens, and therefore it is extremely likely that all ravens are black, that is, that H is true. 'On the other hand', he goes on to argue, 'if there are ravens, then there is a reasonable chance that they are of a variety of colours. Therefore, if I were to discover that even a black raven exists I would consider H to be less probable than it was initially.'
This, according to Good, is as close as one can reasonably expect to get to a condition of perfect ignorance, and it appears that Nicod's condition is still false. 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^{[26]}.
Maher's argument considers a universe of exactly two objects, each of which is very unlikely to be a raven (a one in a thousand chance) and reasonably unlikely to be black (a one in ten chance). Using Carnap's formula for induction, he finds that the probability that all ravens are black decreases from 0.9985 to 0.8995 when it is discovered that one of the two objects is a black raven.
Maher concludes that not only is the paradoxical conclusion true, but that Nicod's criterion is false in the absence of background knowledge (except for the knowledge that the number of objects in the universe is two and that ravens are less likely than black things).
[edit] Distinguished Predicates
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 can be illustrated with Nelson Goodman's example of the predicate grue. An object is grue if it is blue before (say) 2010 and green afterwards. Clearly, we expect objects which were blue before 2010 to remain blue afterwards, but we do not expect the objects which were found to be grue before 2010 to be grue afterwards. Quine's explanation is that "blue" is a natural kind; a privileged predicate which can be used for induction, while "grue" is not a natural kind and using induction with it leads to error.
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 "nonraven". The paradox arises, according to this resolution, because we implicitly interpret Nicod's criterion as applying to all predicates when in fact it only applies to natural kinds.
Another approach which favours specific predicates over others was taken by Hintikka^{[16]}. Hintikka was motivated to find a Bayesian approach to the paradox which did not make use of knowledge about the relative frequencies of ravens and black things. Arguments concerning relative frequencies, he contends, cannot always account for the perceived irrelevance of evidence consisting of observations of objects of type A for the purposes of learning about objects of type notA.
His argument can be illustrated by rephrasing the paradox using predicates other than "raven" and "black". For example, "All men are tall" is equivalent to "All short people are women", and so observing that a randomly selected person is a short woman should provide evidence that all men are tall. 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: "... 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."
His solution is to introduce an order into the set of predicates. When the logical system is equipped with this order, it is possible to restrict the scope of a generalization such as "All ravens are black" so that it applies to ravens only and not to nonblack things, since the order privileges ravens over nonblack things. As he puts it:
 If we are justified in assuming that the scope of the generalization 'All ravens are black' can be restricted to ravens, then this means that we have some outside information which we can rely on concerning the factual situation. The paradox arises from the fact that this information, which colors our spontaneous view of the situation, is not incorporated in the usual treatments of the inductive situation.^{[28]}
[edit] Proposed Resolutions which Reject the Equivalence Condition
[edit] Selective Confirmation
Scheffler and Goodman^{[29]} took an approach to the paradox which incorporates Karl Popper's view that scientific hypotheses are never really confirmed, only falsified.
The approach begins by noting that the observation of a black raven does not prove that "All ravens are black" but it falsifies the contrary hypothesis, "No ravens are black". A nonblack nonraven, on the other hand, is consistent with both "All ravens are black" and with "No ravens are black". As the authors put it:
 ... the statement that all ravens are black is not merely satisfied by evidence of a black raven but is favored by such evidence, since a black raven disconfirms the contrary statement that all ravens are not black, i.e. satisfies its denial. A black raven, in other words, satisfies the hypothesis that all ravens are black rather than not: it thus selectively confirms that all ravens are black.
Selective confirmation violates the equivalence condition since a black raven selectively confirms "All ravens are black" but not "All nonblack things are nonravens".
[edit] Probabilistic or NonProbabilistic Induction
Scheffler and Goodman's concept of selective confirmation is an example of an interpretation of "provides evidence in favor of" which does not coincide with "increase the probability of". This must be a general feature of all resolutions which reject the equivalence condition, since logically equivalent propositions must always have the same probability.
It is impossible for the observation of a black raven to increase the probability of the proposition "All ravens are black" without causing exactly the same change to the probability that "All nonblack things are nonravens". If an observation inductively supports the former but not the latter, then "inductively support" must refer to something other than changes in the probabilities of propositions. A possible loophole is to interpret "All" as "Nearly all"  "Nearly all ravens are black" is not equivalent to "Nearly all nonblack things are nonravens", and these propositions can have very different probabilities^{[30]}.
This raises the broader question of the relation of probability theory to inductive reasoning. Karl Popper argued that probability theory alone cannot account for induction. His argument involves splitting a hypothesis, H, into a part which is deductively entailed by the evidence, E, and another part. This can be done in two ways.
First, consider the splitting^{[31]}:
where A, B and C are probabilistically independent: and so on. The condition which is necessary for such a splitting of H and E to be possible is P(H  E) > P(H), that is, that H is probabilistically supported by E.
Popper's observation is that the part, B, of H which receives support from E actually follows deductively from E, while the part of H which does not follow deductively from E receives no support at all from E  that is, P(A  E) = P(A).
Second, the splitting^{[32]}:
separates H into , which as Popper says, "is the logically strongest part of H (or of the content of H) that follows [deductively] from E," and , which, he says, "contains all of H that goes beyond E." He continues:
 Does E, in this case, provide any support for the factor , which in the presence of E is alone needed to obtain H? The answer is: No. It never does. Indeed, E countersupports unless either P(H  E) = 1 or P(E) = 1 (which are possibilities of no interest). ...
 This result is completely devastating to the inductive interpretation of the calculus of probability. All probabilistic support is purely deductive: that part of a hypothesis that is not deductively entailed by the evidence is always strongly countersupported by the evidence ... There is such a thing as probabilistic support; there might even be such a thing as inductive support (though we hardly think so). But the calculus of probability reveals that probabilistic support cannot be inductive support.
[edit] The Orthodox Approach
The orthodox NeymanPearson theory of hypothesis testing considers how to decide whether to accept or reject a hypothesis, rather than what probability to assign to the hypothesis. From this point of view, the hypothesis that "All ravens are black" is not accepted gradually, as its probability increases towards one when more and more observations are made, but is accepted in a single action as the result of evaluating the data which has already been collected. 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.^{[33]}
According to this approach, it is not necessary to assign any value to the probability of a hypothesis, although one must certainly take into account the probability of the data given the hypothesis, or given a competing hypothesis, when deciding whether to accept or to reject. The acceptance or rejection of a hypothesis carries with it the risk of error.
This contrasts with the Bayesian approach, which requires that the hypothesis be assigned a prior probability, which is revised in the light of the observed data to obtain the final probability of the hypothesis. Within the Bayesian framework there is no risk of error since hypotheses are not accepted or rejected; instead they are assigned probabilities.
An analysis of the paradox from the orthodox point of view has been performed, and leads to, among other insights, a rejection of the equivalence condition:
 It seems obvious that one cannot both accept the hypothesis that all P's are Q and also reject the contrapositive, i.e. that all nonQ's are nonP. Yet it is easy to see that on the NeymanPearson theory of testing, a test of "All P's are Q" is not necessarily a test of "All nonQ's are nonP" or vice versa. A test of "All P's are Q" requires reference to some alternative statistical hypothesis of the form r of all P's are Q, 0 < r < 1, whereas a test of "All nonQ's are nonP" requires reference to some statistical alternative of the form r of all nonQ's are nonP, 0 < r < 1. But these two sets of possible alternatives are different ... Thus one could have a test of H without having a test of its contrapositive.^{[34]}
[edit] Rejecting Material Implication
The following propositions all imply one another: "Every object is either black or not a raven", "Every Raven is black", and "Every nonblack object is a nonraven." They are therefore, by definition, logically equivalent. However, the three propositions have different domains: the first proposition says something about "Every object", while the second says something about "Every raven".
The first proposition is the only one whose domain is unrestricted ("all objects"), so this is the only one which can be expressed in first order logic. It is logically equivalent to:
and also to
where indicates the material conditional, according to which "If A then B" can be understood to mean "B or ".
It has been argued by several authors that material implication does not fully capture the meaning of "If A then B" (see the paradoxes of material implication). "For every object, x, x is either black or not a raven" is true when there are no ravens. It is because of this that "All ravens are black" is regarded as true when there are no ravens. Furthermore, the arguments which Good and Maher used to criticize Nicod's criterion (see Good's Baby, above) relied on this fact  that "All ravens are black" is highly probable when it is highly probable that there are no ravens.
Some approaches to the paradox have sought to find other ways of interpreting "If A then B" and "All A are B" which would eliminate the perceived equivalence between "All ravens are black" and "All nonblack things are nonravens."
One such approach involves introducing a manyvalued logic according to which "If A then B" has the truthvalue I, meaning "Indeterminate" or "Inappropriate" when A is false^{[35]}. In such a system, contraposition is not automatically allowed: "If A then B" is not equivalent to "If then ". Consequently, "All ravens are black" is not equivalent to "All nonblack things are nonravens".
In this system, when contraposition occurs, the modality of the conditional involved changes from the indicative ("If that piece of butter has been heated to 32 C then it has melted") to the counterfactual ("If that piece of butter had been heated to 32 C then it would have melted"). According to this argument, this removes the alleged equivalence which is necessary to conclude that yellow cows can inform us about ravens:
 In proper grammatical usage, a contrapositive argument ought not to be stated entirely in the indicative. Thus:

 From the fact that if this match is scratched it will light, it follows that if it does not light it was not scratched.
 is awkward. We should say:

 From the fact that if this match is scratched it will light, it follows that if it were not to light it would not have been scratched. ...
 One might wonder what effect this interpretation of the Law of Contraposition has on Hempel's paradox of confirmation. "If a is a raven then a is black" is equivalent to "If a were not black then a would not be a raven". Therefore whatever confirms the latter should also, by the Equivalence Condition, confirm the former. True, but yellow cows still cannot figure into the confirmation of "All ravens are black" because, in science, confirmation is accomplished by prediction, and predictions are properly stated in the indicative mood. It is senseless to ask what confirms a counterfactual.^{[36]}
[edit] Differing Results of Accepting the Hypotheses
Several commentators have observed that the propositions "All ravens are black" and "All nonblack things are nonravens" suggest different procedures for testing the hypotheses. E.g. Good writes^{[37]}:
 As propositions the two statements are logically equivalent. But they have a different psychological effect on the experimenter. If he is asked to test whether all ravens are black he will look for a raven and then decide whether it is black. But if he is asked to test whether all nonblack things are nonravens he may look for a nonblack object and then decide whether it is a raven.
More recently, it has been suggested that "All ravens are black" and "All nonblack things are nonravens" can have different effects when accepted^{[38]}. The argument considers situations in which the total numbers or prevalences of ravens and black objects are unknown, but estimated. When the hypothesis "All ravens are black" is accepted, according to the argument, the estimated number of black objects increases, while the estimated number of ravens does not change.
It can be illustrated by considering the situation of two people who have identical information regarding ravens and black objects, and who have identical estimates of the numbers of ravens and black objects. For concreteness, suppose that there are 100 objects overall, and, according to the information available to the people involved, each object is just as likely to be a nonraven as it is to be a raven, and just as likely to be black as it is to be nonblack:
and the propositions are independent for different objects a, b and so on. Then the estimated number of ravens is 50; the estimated number of black things is 50; the estimated number of black ravens is 25, and the estimated number of nonblack ravens (counterexamples to the hypotheses) is 25.
One of the people performs a statistical test (e.g. a NeymanPearson test or the comparison of the accumulated weight of evidence to a threshold) of the hypothesis that "All ravens are black", while the other tests the hypothesis that "All nonblack objects are nonravens". For simplicity, suppose that the evidence used for the test has nothing to do with the collection of 100 objects dealt with here. If the first person accepts the hypothesis that "All ravens are black" then, according to the argument, about 50 objects whose colors were previously in doubt (the ravens) are now thought to be black, while nothing different is thought about the remaining objects (the nonravens). Consequently, he should estimate the number of black ravens at 50, the number of black nonravens at 25 and the number of nonblack nonravens at 25. By specifying these changes, this argument explicitly restricts the domain of "All ravens are black" to ravens.
On the other hand, if the second person accepts the hypothesis that "All nonblack objects are nonravens", then the approximately 50 nonblack objects about which it was uncertain whether each was a raven, will be thought to be nonravens. At the same time, nothing different will be thought about the approximately 50 remaining objects (the black objects). Consequently, he should estimate the number of black ravens at 25, the number of black nonravens at 25 and the number of nonblack nonravens at 50. According to this argument, since the two people disagree about their estimates after they have accepted the different hypotheses, accepting "All ravens are black" is not equivalent to accepting "All nonblack things are nonravens"; accepting the former means estimating more things to be black, while accepting the latter involves estimating more things to be nonravens. Correspondingly, the argument goes, the former requires as evidence ravens which turn out to be black and the latter requires nonblack things which turn out to be nonravens^{[39]}.
[edit] Existential Presuppositions
A number of authors have argued that propositions of the form "All A are B" presuppose that there are objects which are A^{[40]}. This analysis has been applied to the raven paradox^{[41]}:
 ... H_{1}: "All ravens are black" and H_{2}: "All nonblack things are nonravens" are not strictly equivalent ... due to their different existential presuppositions. Moreover, although H_{1} and H_{2} describe the same regularity  the nonexistence of nonblack ravens  they have different logical forms. The two hypotheses have different senses and incorporate different procedures for testing the regularity they describe.
A modified logic can take account of existential presuppositions using the presuppositional operator, '*'. For example,
can denote "All ravens are black" while indicating that it is ravens and not nonblack objects which are presupposed to exist in this example.
 ... the logical form of each hypothesis distinguishes it with respect to its recommended type of supporting evidence: the possibly true substitution instances of each hypothesis relate to different types of objects. The fact that the two hypotheses incorporate different kinds of testing procedures is expressed in the formal language by prefixing the operator '*' to a different predicate. The presuppositional operator thus serves as a relevance operator as well. It is prefixed to the predicate 'x is a raven' in H_{1} because the objects relevant to the testing procedure incorporated in "All raven are black" include only ravens; it is prefixed to the predicate 'x is nonblack', in H_{2}, because the objects relevant to the testing procedure incorporated in "All nonblack things are nonravens" include only nonblack things. ... Using Fregean terms: whenever their presuppositions hold, the two hypotheses have the same referent (truthvalue), but different senses; that is, they express two different ways to determine that truthvalue^{[42]}
[edit] Notes
 ^ ^{a} ^{b} Hempel, CG (1945) Studies in the Logic of Confirmation I. Mind Vol 54, No. 213 p.1 JSTOR
 ^ Hempel, CG (1945) Studies in the Logic of Confirmation II. Mind Vol 54, No. 214 p.97 JSTOR
 ^ Maher, P (1999) Inductive Logic and the Ravens Paradox, Philosphoy of Science, 66, p.50 JSTOR
 ^ Good, IJ (1960) The Paradox of Confirmation, The British Journal for the Philosophy of Science, Vol. 11, No. 42, 145149 JSTOR
 ^ Fitelson, B and Hawthorne, J (2006) How Bayesian Confirmation Theory Handles the Paradox of the Ravens, in Probability in Science, Chicago: Open Court Link
 ^ Alexander, HG (1958) The Paradoxes of Confirmation, The British Journal for the Philosophy of Science, Vol. 9, No. 35, P. 227 JSTOR
 ^ HosaissonLindenbaum, J (1940) On Confirmation, The Journal of Symbolic Logic, Vol. 5, No. 4, p. 133 JSTOR
 ^ Note: Good used "crow" instead of "raven", but "raven" has been used here throughout for consistency.
 ^ Chihara, (1981) Some Problems for Bayesian Confirmation Theory, British Journal for the Philosophy of Science, Vol. 38, No. 4 LINK
 ^ Earman, 1992 Bayes or Bust? A Critical Examination of Bayesian Confirmation Theory, MIT Press, Cambridge, MA.
 ^ Eells, 1982 Rational Decision and Causality. New York: Cambridge University Press
 ^ Gibson, 1969 On Ravens and Relevance and a Likelihood Solution of the Paradox of Confirmation, LINK
 ^ HosaissonLindenbaum 1940
 ^ Howson, Urbach, 1993 Scientific Reasoning: The Bayesian Approach, Open Court Publishing Company
 ^ Mackie, 1963 The Paradox of Confirmation, Brit. J. Phil. Sci. Vol. 13, No. 52, p. 265 LINK
 ^ ^{a} ^{b} Hintikka, 1969
 ^ Humburg 1986, The solution of Hempel's raven paradox in Rudolf Carnap's system of inductive logic, Erkenntnis, Vol. 24, No. 1, pp
 ^ Maher 1999
 ^ Fitelson 2006
 ^ Vranas (2002) Hempel's Raven Paradox: A Lacuna in the Standard Bayesian Solution LINK
 ^ ^{a} ^{b} Maher, 1999
 ^ ^{a} ^{b} Fitelson, 2006
 ^ Good 1967, The White Shoe is a Red Herring, British Journal for the Philosophy of Science, Vol. 17, No. 4, p322 JSTOR
 ^ Hempel 1967, The White Shoe  No Red Herring, The British Journal for the Philosophy of Science, Vol. 18, No. 3, p. 239 JSTOR
 ^ Good 1968, The White Shoe qua Red Herring is Pink, The British Journal for the Philosophy of Science, Vol. 19, No. 2, p. 156 JSTOR
 ^ Maher 2004, Probability Captures the Logic of Scientific Confirmation LINK
 ^ Quine, WV (1969) Natural Kinds, in Ontological Relativity and other Essays. New York:Columbia university Press, p.114
 ^ Hintikka J. 1969, Inductive Independence and the Paradoxes of Confirmation LINK
 ^ Scheffler I, Goodman NJ, Selective Confirmation and the Ravens, Journal of Philosophy, Vol. 69, No. 3, 1972 JSTOR
 ^ Gaifman, H (1979) Subjective Probability, Natural Predicates and Hempel's Ravens, Erkenntnis, Vol. 14, p. 105 Springer
 ^ Popper, K. Realism and the Aim of Science, Routlege, 1992, p. 325
 ^ Popper K, Miller D, (1983) A Proof of the Impossibility of Inductive Probability, Nature, Vol. 302, p. 687 Link
 ^ Neyman J, Pearson ES (1933) On the Problem of the Most Efficient Tests of Statistical Hypotheses, Phil. Transactions of the Royal Society of London. Series A, Vol. 231, p289 JSTOR
 ^ Giere, RN (1970) An Orthodox Statistical Resolution of the Paradox of Confirmation, Philosophy of Science, Vol. 37, No. 3, p.354 JSTOR
 ^ Farrell RJ (1979) Material Implication, Confirmation and Counterfactuals LINK
 ^ Farrell (1979)
 ^ Good (1960)
 ^ O'Flanagan (2008) Judgment LINK
 ^ O'Flanagan (2008)
 ^ Strawson PF (1952) Introduction to Logical Theory, methuan & Co. London, John Wiley & Sons, New York
 ^ Cohen Y (1987) Ravens and Relevance, Erkenntnis LINK
 ^ Cohen (1987)
[edit] References
 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, 139156, 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, 122143, 1943.
 Hempel, C. G. Studies in Logic and Confirmation. Mind 54, 126, 1945.
 Hempel, C. G. Studies in Logic and Confirmation. II. Mind 54, 97121, 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. 145183.
 Whiteley, C. H. Hempel's Paradoxes of Confirmation. Mind 55, 156158, 1945.