Editing Raven paradox (section)

===Disputing the induction from positive instances===
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).

====Red herring====
Good<ref>{{cite journal |last1=Good |first1=I. J. |year=1967 |title=The White Shoe is a Red Herring |journal=British Journal for the Philosophy of Science |volume=17 |issue=4 |page=322 |jstor=686774 |doi=10.1093/bjps/17.4.322 }}</ref> gives an example of background knowledge with respect to which the observation of a black raven ''decreases'' the probability that all ravens are black:

{{quote|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 non-black 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."<ref>{{cite journal | last1 = Hempel | year = 1967 | title = The White Shoe—No Red Herring | journal = The British Journal for the Philosophy of Science | volume = 18 | issue = 3| pages = 239–240| jstor=686596 | doi=10.1093/bjps/18.3.239}}</ref>

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.

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 our background knowledge itself is the red herring, and that we should consider induction with respect to a condition of perfect ignorance.

====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:<ref>{{cite journal |last1=Good |year=1968 |title=The White Shoe qua Red Herring is Pink |journal=The British Journal for the Philosophy of Science |volume=19 |issue=2 |pages=156–157 |jstor=686795 |doi=10.1093/bjps/19.2.156 |first1=I. J. }}</ref>

{{quote|...&nbsp;imagine an infinitely intelligent newborn baby having built-in 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 <math>H</math> 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 <math>H</math> 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.<ref>{{cite book | editor-first=Christopher |editor-last=Hitchcock| title=Contemporary Debates in the Philosophy of Science |publisher=Blackwell |first=Patrick |last=Maher |year=2004 |chapter=Probability Captures the Logic of Scientific Confirmation |pages=69–93 |chapter-url=http://patrick.maher1.net/pctl.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://patrick.maher1.net/pctl.pdf |archive-date=2022-10-09 |url-status=live}}</ref>

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).

====Distinguished predicates====
Quine<ref>{{cite book|title=Essays in Honor of Carl G. Hempel|author-link=Willard Van Orman Quine|first=Willard Van Orman |last=Quine|publisher=D. Reidel|year=1970|editor-first=Nicholas |editor-last=Rescher|location=Dordrecht|pages=41–56|chapter=Natural Kinds|display-editors=etal|chapter-url=http://fitelson.org/confirmation/quine_nk.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://fitelson.org/confirmation/quine_nk.pdf |archive-date=2022-10-09 |url-status=live}} Reprinted in: {{cite book|title=Ontological Relativity and other Essays|last=Quine|first=W. V.|author-link=Willard Van Orman Quine|publisher=Columbia University Press|year=1969|location=New York|page=114|contribution=Natural Kinds}}<!---chapter=5---><!---According to Quine's foreword to the latter book (p.6 in the german translation), the reprint was issued earlier than the original.---></ref> argued that the solution to the paradox lies in the recognition that certain [[predicate (mathematical logic)|predicate]]s, which he called [[natural kind]]s, have a distinguished status with respect to induction. This can be illustrated with [[Nelson Goodman]]'s example of the predicate [[Grue and Bleen|grue]]. An object is grue if it is blue before (say) {{YEAR|{{TODAY}}}} and green afterwards. Clearly, we expect objects that were blue before {{YEAR|{{TODAY}}}} to remain blue afterwards, but we do not expect the objects that were found to be grue before {{YEAR|{{TODAY}}}} to be blue after {{YEAR|{{TODAY}}}}, since after {{YEAR|{{TODAY}}}} they would be green. Quine's explanation is that "blue" is a natural kind; a privileged predicate we can use 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 "non-raven". 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.<ref name=Hintikka1970/> 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 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 as follows: "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."<ref name=Hintikka1970/>

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 non-black things, since the order privileges ravens over non-black things. As he puts it:

{{quote|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.<ref name=Hintikka1970/>}}