Editing Raven paradox (section)

====Selective confirmation====
Scheffler and Goodman<ref>{{cite journal | last1 = Scheffler | first1 = I | last2 = Goodman | first2 = N. J. | title = Selective Confirmation and the Ravens | journal = Journal of Philosophy | volume = 69 | issue = 3| pages = 78–83 | year = 1972 | jstor=2024647| doi = 10.2307/2024647 }}</ref> took an approach to the paradox that incorporates [[Karl Popper]]'s view that scientific hypotheses are never really confirmed, only falsified.

The approach begins by noting that the observation of a black raven does not prove that "All ravens are black" but it falsifies the contrary hypothesis, "No ravens are black". A non-black non-raven, on the other hand, is consistent with both "All ravens are black" and with "No ravens are black". As the authors put it:

{{quote|... the statement that all ravens are black is not merely ''satisfied'' by evidence of a black raven but is ''favored'' by such evidence, since a black raven disconfirms the contrary statement that all ravens are not black, i.e. satisfies its denial. A black raven, in other words, satisfies the hypothesis ''that all ravens are black rather than not:'' it thus selectively confirms ''that all ravens are black''.}}

Selective confirmation violates the equivalence condition since a black raven selectively confirms "All ravens are black" but not "All non-black things are non-ravens".

=====Probabilistic or non-probabilistic induction=====
Scheffler and Goodman's concept of selective confirmation is an example of an interpretation of "provides evidence in favor of..." which does not coincide with "increase the probability of..." This must be a general feature of all resolutions that reject the equivalence condition, since logically equivalent propositions must always have the same probability.

It is impossible for the observation of a black raven to increase the probability of the proposition "All ravens are black" without causing exactly the same change to the probability that "All non-black things are non-ravens". If an observation inductively supports the former but not the latter, then "inductively support" must refer to something other than changes in the probabilities of propositions. A possible loophole is to interpret "All" as "Nearly all" – "Nearly all ravens are black" is not equivalent to "Nearly all non-black things are non-ravens", and these propositions can have very different probabilities.<ref>{{cite journal |last=Gaifman |first=H. |year=1979 |title=Subjective Probability, Natural Predicates and Hempel's Ravens |journal=[[Erkenntnis]] |volume=14 |issue=2 |pages=105–147 |doi=10.1007/BF00196729 |s2cid=189891124 }}</ref>

This raises the broader question of the relation of probability theory to inductive reasoning. [[Karl Popper]] argued that probability theory alone cannot account for induction. His argument involves splitting a hypothesis, <math>H</math>, into a part that is deductively entailed by the evidence, <math>E</math>, and another part. This can be done in two ways.

First, consider the splitting:<ref>Popper, K. ''Realism and the Aim of Science'', Routledge, 1992, p. 325</ref>

<math display="block">H=A\ and\ B \ \ \ \ \ \ E=B\ and\ C</math>

where <math>A</math>, <math>B</math> and <math>C</math> are probabilistically independent: <math>P(A\ and\ B)=P(A)P(B)</math> and so on. The condition that is necessary for such a splitting of H and E to be possible is <math>P(H|E)>P(H)</math>, that is, that <math>H</math> is probabilistically supported by <math>E</math>.

Popper's observation is that the part, <math>B</math>, of <math>H</math> that receives support from <math>E</math> actually follows deductively from <math>E</math>, while the part of <math>H</math> that does not follow deductively from <math>E</math> receives no support at all from <math>E</math> – that is, <math>P(A|E)=P(A)</math>.

Second, the splitting:<ref>{{cite journal | last1 = Popper | first1 = K. | last2 = Miller | first2 = D. | year = 1983 | title = A Proof of the Impossibility of Inductive Probability | journal = Nature | volume = 302 | issue = 5910| page = 687 | doi=10.1038/302687a0| bibcode = 1983Natur.302..687P | s2cid = 4317588 }}</ref>

<math display="block">H=(H\ or\ E)\ and\ (H\ or\ \overline{E})</math>

separates <math>H</math> into <math>(H\ or\ E)</math>, which as Popper says, "is the logically strongest part of <math>H</math> (or of the content of <math>H</math>) that follows [deductively] from <math>E</math>", and <math>(H\ or\ \overline{E})</math>, which, he says, "contains all of <math>H</math> that goes beyond <math>E</math>". He continues:

{{quote|
Does <math>E</math>, in this case, provide any support for the factor <math>(H\ or\ \overline{E})</math>, which in the presence of <math>E</math> is alone needed to obtain <math>H</math>? The answer is: No. It never does. Indeed, <math>E</math> countersupports <math>(H\ or\ \overline{E})</math> unless either <math>P(H|E)=1</math> or <math>P(E)=1</math> (which are possibilities of no interest). ...

This result is completely devastating to the inductive interpretation of the calculus of probability. All probabilistic support is purely deductive: that part of a hypothesis that is not deductively entailed by the evidence is always strongly countersupported by the evidence ... There is such a thing as probabilistic support; there might even be such a thing as inductive support (though we hardly think so). But the calculus of probability reveals that probabilistic support cannot be inductive support.}}