Editing Raven paradox (section)

===Accepting non-ravens as relevant===
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'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:<ref name="JSTOR"/>{{rp|19–20}}

{{quotation|
This result would confirm the assertion, "Whatever does not burn yellow is not 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.}}

====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<ref name=Good1960>{{cite journal|jstor=685588|title=The Paradox of Confirmation|journal=The British Journal for the Philosophy of Science|first=I. J.|last=Good|date=1960|volume=11|issue=42|pages=145–149|doi=10.1093/bjps/XI.42.145-b}}</ref> is perhaps the best known, and variations of the argument have been popular ever since,<ref name=Fitelson&Hawthorne2006>{{cite book |last1=Fitelson |first1=Branden |last2=Hawthorne |first2=James |date=2010 |chapter=How Bayesian confirmation theory handles the paradox of the ravens |editor1-last=Eells |editor1-first=Ellery |editor2-last=Fetzer |editor2-first=James H. |title=The place of probability in science: in honor of Ellery Eells (1953–2006) |series=Boston studies in the philosophy of science |volume=284 |location=Dordrecht; New York |publisher=Springer |isbn=9789048136148 |oclc=436266507 |doi=10.1007/978-90-481-3615-5_11 |chapter-url=http://fitelson.org/ravens.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://fitelson.org/ravens.pdf |archive-date=2022-10-09 |url-status=live}}</ref> although it had been presented in 1958<ref>{{cite journal | last1 = Alexander | first1 = H. G. | year = 1958 | title = The Paradoxes of Confirmation | jstor = 685654| journal = The British Journal for the Philosophy of Science | volume = 9 | issue = 35| pages = 227–233 | doi=10.1093/bjps/ix.35.227| s2cid = 120300549 }}</ref> and early forms of the argument appeared as early as 1940.<ref name=Hosiasson-Lindenbaum1940>{{cite journal |author=Janina Hosiasson-Lindenbaum |year=1940 |title=On Confirmation |journal=The Journal of Symbolic Logic |volume=5 |issue=4 |pages=133–148 |url=http://fitelson.org/confirmation/lindenbaum.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://fitelson.org/confirmation/lindenbaum.pdf |archive-date=2022-10-09 |url-status=live |doi=10.2307/2268173|jstor=2268173 |s2cid=195347283 |author-link=Janina Hosiasson-Lindenbaum }}</ref>

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:

{{quote|... suppose that there are <math>N</math> objects that might be seen at any moment, of which <math>r</math> are ravens and <math>b</math> are black, and that the <math>N</math> objects each have probability <math>\tfrac{1}{N}</math> of being seen. Let <math>H_i</math> be the hypothesis that there are <math>i</math> non-black ravens, and suppose that the hypotheses <math>H_1, H_2, ... ,H_r</math> are initially equiprobable. Then, if we happen to see a black raven, the Bayes factor in favour of <math>H_0</math> is

<math display="block">\tfrac{r}{N} \Big / \text{average} \left( \tfrac{r-1}{N},\tfrac{r-2}{N}, ...\ ,\tfrac{1}{N}\right) \ = \ \tfrac{2r}{r-1}</math>

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

<math display="block">\begin{array}{c}
\tfrac{N-b}{N} \Big / \text{average} \left( \tfrac{N-b-1}{N},\tfrac{N-b-2}{N}, ...\ ,\max(0,\tfrac{N-b-r}{N})\right) \\
\ = \ \frac{N-b}{\max\left(N-b-\tfrac{r}{2}-\tfrac12\ , \ \tfrac12(N-b-1)\right)}
\end{array}</math>

and this exceeds unity by only about <math>r/(2N-2b)</math> if <math>N-b</math> is large compared to <math>r</math>. 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 non-black objects.<ref>Note: Good used "crow" instead of "raven", but "raven" has been used here throughout for consistency.</ref>}}

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 [[Charles Chihara | Chihara]]<ref>{{cite journal | last1 = Chihara | year = 1987 | title = Some Problems for Bayesian Confirmation Theory | url = http://bjps.oxfordjournals.org/cgi/reprint/38/4/551 | journal = British Journal for the Philosophy of Science | volume = 38 | issue = 4 | page = 551 | doi=10.1093/bjps/38.4.551}}</ref> 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 (some of which accept !PC and instead reject NC) include Earman,<ref>Earman, 1992 ''Bayes or Bust? A Critical Examination of Bayesian Confirmation Theory'', Cambridge, Massachusetts: MIT Press.</ref> Eells,<ref>Eells, 1982 ''Rational Decision and Causality''. New York: Cambridge University Press.</ref> Gibson,<ref>Gibson, 1969 [https://www.jstor.org/stable/686720 "On Ravens and Relevance and a Likelihood Solution of the Paradox of Confirmation"]</ref> [[Janina Hosiasson-Lindenbaum|Hosiasson-Lindenbaum]],<ref name=Hosiasson-Lindenbaum1940/> Howson and Urbach,<ref>Howson, Urbach, 1993 ''Scientific Reasoning: The Bayesian Approach'', Open Court Publishing Company</ref> Mackie,<ref>{{cite journal | last1 = Mackie | year = 1963 | title = The Paradox of Confirmation | url = http://bjps.oxfordjournals.org/cgi/content/citation/XIII/52/265 | journal = The British Journal for the Philosophy of Science | volume = 13 | issue = 52| page = 265 | doi=10.1093/bjps/xiii.52.265}}</ref> and Hintikka,<ref name=Hintikka1970>{{cite book |last=Hintikka |first=Jaakko |author-link=Jaakko Hintikka |date=1970 |chapter=Inductive independence and the paradoxes of confirmation |editor-last=Rescher |editor-first=Nicholas |title=Essays in honor of Carl G. Hempel: a tribute on the occasion of his sixty-fifth birthday |series=Synthese library |location=Dordrecht |publisher=[[D. Reidel]] |pages=[https://books.google.com/books?id=pWtPcRwuacAC&pg=PA24 24–46] |oclc=83854 |doi=10.1007/978-94-017-1466-2_3 |isbn=978-90-481-8332-6 |chapter-url=https://books.google.com/books?id=pWtPcRwuacAC&pg=PA24}}</ref> 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,<ref>Humburg 1986, "The solution of Hempel's raven paradox in Rudolf Carnap's system of inductive logic", ''[[Erkenntnis]]'', Vol. 24, No. 1, pp</ref> Maher,<ref name=Maher1999/> and Fitelson & Hawthorne.<ref name=Fitelson&Hawthorne2006/> Vranas<ref>Vranas (2002) [http://philsci-archive.pitt.edu/archive/00000688/00/hempelacuna.doc "Hempel's Raven Paradox: A Lacuna in the Standard Bayesian Solution"] {{Webarchive|url=https://web.archive.org/web/20100712042954/http://philsci-archive.pitt.edu/archive/00000688/00/hempelacuna.doc |date=2010-07-12 }}</ref> introduced the term "Standard Bayesian Solution" to avoid confusion.

====Carnap approach====
Maher<ref name=Maher1999/> accepts the paradoxical conclusion, and refines it:

{{Quote|A non-raven (of whatever color) confirms that all ravens are black because

{{unbulleted list|item_style=margin-left: 1.5em; text-indent: -1.5em|(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.}}}}

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. According to Carnap's theory, the posterior probability, <math>P(Fa|E)</math>, that an object, <math>a</math>, will have a predicate, <math>F</math>, after the evidence <math>E</math> has been observed, is:

<math display="block">P(Fa|E) \ = \ \frac{n_F+\lambda P(Fa)}{n+\lambda}</math>

where <math>P(Fa)</math> is the initial probability that <math>a</math> has the predicate <math>F</math>; <math>n</math> is the number of objects that have been examined (according to the available evidence <math>E</math>); <math>n_F</math> is the number of examined objects that turned out to have the predicate <math>F</math>, and <math>\lambda</math> is a constant that measures resistance to generalization.

If <math>\lambda</math> is close to zero, <math>P(Fa|E)</math> will be very close to one after a single observation of an object that turned out to have the predicate <math>F</math>, while if <math>\lambda</math> is much larger than <math>n</math>, <math>P(Fa|E)</math> will be very close to <math>P(Fa)</math> regardless of the fraction of observed objects that had the predicate <math>F</math>.

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.

====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<ref name=Maher1999/> 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 [[Logical conjunction|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 (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. 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 [i.e. the Standard Bayesian one] cannot be correct."

Fitelson & Hawthorne<ref name=Fitelson&Hawthorne2006/> 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 <math>a</math> is an object selected at random, <math>Ba</math> is the proposition that the object is black, and <math>Ra</math> is the proposition that the object is a raven, then the condition:

<math display="block">\frac{P(\overline{Ba}|\overline{H})}{P(Ra|\overline{H})} \ - \ P(\overline{Ba}|Ra\overline{H}) \  \geq \ P(Ba|Ra\overline{H}) \frac{P(\overline{Ba}|H)}{P(Ra|H)}</math>

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 <math>-\log P(Ba|Ra\overline{H})</math>. This is equal to the amount of additional information (in bits, if the base of the logarithm is 2) 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<ref name=Fitelson&Hawthorne2006/> explain that:

{{quote|Under normal circumstances, <math>p=P(Ba|Ra\overline{H})</math> may be somewhere around 0.9 or 0.95; so <math>1/p</math> 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 non-black non-raven. However, under plausible conditions it can be shown that a sequence of <math>n</math> instances (i.e. of n black ravens, as compared to n non-black non-ravens) yields a ratio of likelihood ratios on the order of <math>(1/p)^n</math>, which blows up significantly for large <math>n</math>.}}

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.