BLACK RAVEN 1 !!INSTALL!!
The raven paradox, also known as Hempel's paradox, Hempel's ravens, or rarely the paradox of indoor ornithology, is a paradox arising from the question of what constitutes evidence for the truth of a statement. Observing objects that are neither black nor ravens may formally increase the likelihood that all ravens are black even though, intuitively, these observations are unrelated.
BLACK RAVEN 1
Given a general statement such as all ravens are black, a form of the same statement that refers to a specific observable instance of the general class would typically be considered to constitute evidence for that general statement. For example,
By the same reasoning, this statement is evidence that (2) if something is not black then it 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 is evidence supporting the notion that all ravens are black. This conclusion seems paradoxical because it implies that information has been gained about ravens by looking at an apple.
Nicod's criterion says that only observations of ravens should affect one's view as to whether all ravens are black. Observing more instances of black ravens should support the view, observing white or coloured ravens should contradict it, and observations of non-ravens should not have any influence.
Realistically, the set of ravens is finite. The set of non-black things is either infinite or beyond human enumeration. To confirm the statement 'All ravens are black', it would be necessary to observe all ravens. This is difficult but possible. To confirm the statement 'All non-black things are non-ravens', it would be necessary to examine all non-black things. This is not possible. Observing a black raven could be considered a finite amount of confirmatory evidence, but observing a non-black non-raven would be an infinitesimal amount of evidence.
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.
Hempel himself accepted the paradoxical conclusion, arguing that the reason the result appears paradoxical is that we possess prior information without which the observation of a non-black non-raven would indeed provide evidence that all ravens are black.
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.
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:
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.
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 shows that, for a large class of possible configurations of background knowledge, the observation of a non-black non-raven provides exactly the same amount of confirmation as the observation of a black raven. The configurations of background knowledge that he considers are those that are provided by a sample proposition, namely a proposition that 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.
Fitelson & Hawthorne examined the conditions under which the observation of a non-black non-raven provides less evidence than the observation of a black raven. They show that, if a \displaystyle a is an object selected at random, B a \displaystyle Ba is the proposition that the object is black, and R a \displaystyle Ra is the proposition that the object is a raven, then the condition:
is sufficient for the observation of a non-black non-raven to provide less evidence than the observation of a black raven. Here, a line over a proposition indicates the logical negation of that proposition.
The authors point out that their analysis is completely consistent with the supposition that a non-black non-raven 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 non-black non-raven provides.
Some approaches for resolving the paradox focus on the inductive step. They dispute whether observation of a particular instance (such as one black raven) is the kind of evidence that necessarily increases confidence in the general hypothesis (such as that ravens are always 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."
The question that 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 that we actually possess regarding ravens and black objects, or with regard to all possible configurations of background information.
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:
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.
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).
Another approach, which favours specific predicates over others, was taken by Hintikka. 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. 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 not-A.
His argument can be illustrated by rephrasing the paradox using predicates other than "raven" and "black". For example, "All menare 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." 041b061a72