Raven paradox
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.
This problem was proposed by the logician Carl Gustav Hempel in the 1940s to illustrate a contradiction between inductive logic and intuition.
Paradox
Hempel describes the paradox in terms of the hypothesis:Via contraposition, this statement is equivalent to:
In all circumstances where is true, is also true—and likewise, in all circumstances where is false, is also false.
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,
is evidence supporting the hypothesis that all ravens are black.
The paradox arises when this same process is applied to statement. On sighting a green apple, one can observe:
By the same reasoning, this statement is evidence that if something is not black then it is not a raven. But since this statement is logically equivalent to 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.
Proposed resolutions
'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.Hempel's equivalence condition states that when a proposition, X, provides evidence in favor of another proposition Y, then X also provides evidence in favor of any proposition that is logically equivalent to Y.
The paradox shows that Nicod's criterion and Hempel's equivalence condition are not mutually consistent. A resolution to the paradox must reject at least one out of:
- negative instances having no influence,
- equivalence condition, or,
- validation by positive instances.
Accepting non-ravens as relevant
Although this conclusion of the paradox seems counter-intuitive, some approaches accept that observations of non-ravens can in fact constitute valid evidence in support for hypotheses about ravens.Hempel's resolution
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.He illustrates this with the example of the generalization "All sodium salts burn yellow", and asks us to consider the observation that occurs when somebody holds a piece of pure ice in a colorless flame that does not turn yellow:
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 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.I. J. Good's presentation of this argument in 1960 is perhaps the best known, and variations of the argument have been popular ever since, although it had been presented in 1958 and early forms of the argument appeared as early as 1940.
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:
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 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, Eells, Gibson, Hosiasson-Lindenbaum, Howson and Urbach, Mackie, and Hintikka, 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, Maher, and Fitelson & Hawthorne. Vranas introduced the term "Standard Bayesian Solution" to avoid confusion.
Carnap approach
Maher accepts the paradoxical conclusion, and refines it:To reach, he appeals to Carnap's theory of inductive probability, which is a way of assigning prior probabilities that naturally implements induction. According to Carnap's theory, the posterior probability,, that an object,, will have a predicate,, after the evidence has been observed, is:
where is the initial probability that has the predicate ; is the number of objects that have been examined ; is the number of examined objects that turned out to have the predicate, and is a constant that measures resistance to generalization.
If is close to zero, will be very close to one after a single observation of an object that turned out to have the predicate, while if is much larger than, will be very close to regardless of the fraction of observed objects that had the predicate.
Using this Carnapian approach, Maher identifies a proposition we intuitively 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 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.
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 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.Maher's proof appears to contradict the result of the Bayesian argument, which was that the observation of a non-black non-raven provides much less evidence than the observation of a black raven. 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 that the total numbers of ravens, non-black objects and/or the total number of objects, are known quantities. Maher comments that, "The reason we think there are more non-black 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 non-black non-raven confirms A just as strongly as a black raven does... Thus my analysis suggests that this response to the paradox cannot be correct."
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 is an object selected at random, is the proposition that the object is black, and 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.
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 non-black non-raven by about. This is equal to the amount of additional information that is provided when a raven of unknown color is discovered to be black, given the hypothesis that not all ravens are black.
Fitelson & Hawthorne explain that:
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.