Editing Raven paradox (section)

====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.