Editing Raven paradox (section)

===Rejections of Hempel's equivalence condition===
Some approaches for the resolution of the paradox reject Hempel's equivalence condition. That is, they may not consider evidence supporting the statement ''all non-black objects are non-ravens'' to necessarily support logically-equivalent statements such as ''all ravens are black''.

====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.}}

====Orthodox approach====
The orthodox [[Type I and type II errors|Neyman–Pearson]] theory of hypothesis testing considers how to decide whether to ''accept'' or ''reject'' a hypothesis, rather than what probability to assign to the hypothesis. From this point of view, the hypothesis that "All ravens are black" is not accepted ''gradually'', as its probability increases towards one when more and more observations are made, but is accepted in a single action as the result of evaluating the data that has already been collected. As Neyman and Pearson put it:

{{quote|Without hoping to know whether each separate hypothesis is true or false, we may search for rules to govern our behaviour with regard to them, in following which we insure that, in the long run of experience, we shall not be too often wrong.<ref>{{cite journal |last1=Neyman |first1=J. |last2=Pearson |first2=E. S. |date=1933 |title=On the Problem of the Most Efficient Tests of Statistical Hypotheses |journal=Philosophical Transactions of the Royal Society A| volume=231|issue=694–706 | page=289 |url=http://www.stats.org.uk/statistical-inference/NeymanPearson1933.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.stats.org.uk/statistical-inference/NeymanPearson1933.pdf |archive-date=2022-10-09 |url-status=live |doi=10.1098/rsta.1933.0009|bibcode=1933RSPTA.231..289N |jstor=91247|doi-access=free }}</ref>}}

According to this approach, it is not necessary to assign any value to the probability of a ''hypothesis'', although one must certainly take into account the probability of the ''data'' given the hypothesis, or given a competing hypothesis, when deciding whether to accept or to reject. The acceptance or rejection of a hypothesis carries with it the risk of [[Type I and type II errors|error]].

This contrasts with the Bayesian approach, which requires that the hypothesis be assigned a prior probability, which is revised in the light of the observed data to obtain the final probability of the hypothesis. Within the Bayesian framework there is no risk of error since hypotheses are not accepted or rejected; instead they are assigned probabilities.

An analysis of the paradox from the orthodox point of view has been performed, and leads to, among other insights, a rejection of the equivalence condition:

{{quote|It seems obvious that one cannot both ''accept'' the hypothesis that all P's are Q and also reject the contrapositive, i.e. that all non-Q's are non-P. Yet it is easy to see that on the Neyman-Pearson theory of testing, a test of "All P's are Q" is ''not'' necessarily a test of "All non-Q's are non-P" or vice versa. A test of "All P's are Q" requires reference to some alternative statistical hypothesis of the form <math>r</math> of all P's are Q, <math>0<r<1</math>, whereas a test of "All non-Q's are non-P" requires reference to some statistical alternative of the form <math>r</math> of all non-Q's are non-P, <math>0<r<1</math>. But these two sets of possible alternatives are different ... Thus one could have a test of <math>H</math> without having a test of its contrapositive.<ref>{{cite journal | last1 = Giere | first1 = R. N. | year = 1970 | title = An Orthodox Statistical Resolution of the Paradox of Confirmation | journal = Philosophy of Science | volume = 37 | issue = 3| pages = 354–362 | jstor=186464 | doi=10.1086/288313| s2cid = 119854130 }}</ref>}}

====Rejecting material implication====
The following propositions all imply one another: "Every object is either black or not a raven", "Every raven is black", and "Every non-black object is a non-raven." They are therefore, by definition, logically equivalent. However, the three propositions have different domains: the first proposition says something about "every object", while the second says something about "every raven".

The first proposition is the only one whose domain of quantification is unrestricted ("all objects"), so this is the only one that can be expressed in [[first-order logic]]. It is logically equivalent to:

<math display="block">\forall\ x, Rx\ \rightarrow\ Bx</math>

and also to

<math display="block">\forall\ x, \overline{Bx}\ \rightarrow\ \overline{Rx}</math>

where <math>\rightarrow</math> indicates the [[material conditional]], according to which "If <math>A</math> then {{nobr|<math>B</math>"}} can be understood to mean {{nobr|"<math>B</math> or <math>\overline{A}</math>".}}

It has been argued by several authors that material implication does not fully capture the meaning of "If <math>A</math> then {{nobr|<math>B</math>"}} (see the [[paradoxes of material implication]]). "For every object, {{nobr|<math>x</math>,}} <math>x</math> is either black or not a raven" is ''true'' when there are no ravens. It is because of this that "All ravens are black" is regarded as true when there are no ravens. Furthermore, the arguments that Good and Maher used to criticize Nicod's criterion (see {{slink||Good's baby}}, above) relied on this fact – that "All ravens are black" is highly probable when it is highly probable that there are no ravens.

To say that all ravens are black in the absence of any ravens is an empty statement. It refers to nothing. "All ravens are white" is equally relevant and true, if this statement is considered to have any truth or relevance.

Some approaches to the paradox have sought to find other ways of interpreting "If <math>A</math> then {{nobr|<math>B</math>"}} and "All <math>A</math> are {{nobr|<math>B</math>,"}} which would eliminate the perceived equivalence between "All ravens are black" and "All non-black things are non-ravens."

One such approach involves introducing a [[many-valued logic]] according to which "If <math>A</math> then {{nobr|<math>B</math>"}} has the [[truth value]] {{nobr|<math>I</math>,}} meaning "Indeterminate" or "Inappropriate" when <math>A</math> is false.<ref name=Farell1979>{{cite journal| last=Farrell |first=R. J.| title=Material Implication, Confirmation and Counterfactuals| journal=Notre Dame Journal of Formal Logic|date=April 1979| volume=20| number=2| pages=383–394| url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ndjfl/1093882546 |doi=10.1305/ndjfl/1093882546| doi-access=free}}</ref> In such a system, [[contraposition]] is not automatically allowed: "If <math>A</math> then {{nobr|<math>B</math>"}} is not equivalent to "If <math>\overline{B}</math> then {{nobr|<math>\overline{A}</math>".}} Consequently, "All ravens are black" is not equivalent to "All non-black things are non-ravens".

In this system, when contraposition occurs, the [[Linguistic modality|modality]] of the conditional involved changes from the [[indicative]] ("If that piece of butter ''has been'' heated to 32&nbsp;°C then it ''has'' melted") to the counterfactual ("If that piece of butter ''had been'' heated to 32&nbsp;°C then it ''would have'' melted"). According to this argument, this removes the alleged equivalence that is necessary to conclude that yellow cows can inform us about ravens:

{{quote|
In proper grammatical usage, a contrapositive argument ought not to be stated entirely in the indicative. Thus:

{{quote|style=font-size:inherit|From the fact that if this match is scratched it will light, it follows that if it does not light it was not scratched.}}

is awkward. We should say:

{{quote|style=font-size:inherit|From the fact that if this match is scratched it will light, it follows that if it ''were'' not to light it ''would'' not have been scratched.&nbsp;...}}

One might wonder what effect this interpretation of the Law of Contraposition has on Hempel's paradox of confirmation. "If <math>a</math> is a raven then <math>a</math> is black" is equivalent to "If <math>a</math> were not black then <math>a</math> would not be a raven". Therefore whatever confirms the latter should also, by the Equivalence Condition, confirm the former. True, but yellow cows still cannot figure into the confirmation of "All ravens are black" because, in science, confirmation is accomplished by prediction, and predictions are properly stated in the indicative mood. It is senseless to ask what confirms a counterfactual.<ref name=Farell1979/>}}

====Differing results of accepting the hypotheses====
Several commentators have observed that the propositions "All ravens are black" and "All non-black things are non-ravens" suggest different procedures for testing the hypotheses. E.g. Good writes:<ref name=Good1960/>

{{quote|As propositions the two statements are logically equivalent. But they have a different psychological effect on the experimenter. If he is asked to test whether all ravens are black he will look for a raven and then decide whether it is black. But if he is asked to test whether all non-black things are non-ravens he may look for a non-black object and then decide whether it is a raven.}}

More recently, it has been suggested that "All ravens are black" and "All non-black things are non-ravens" can have different effects when ''accepted''.<ref name=O'Flanagan2008>{{cite arXiv| author=Ruadhan O'Flanagan |date=Feb 2008 |title=Judgment |eprint=0712.4402 |class=math.PR }}</ref> The argument considers situations in which the total numbers or prevalences of ravens and black objects are unknown, but estimated. When the hypothesis "All ravens are black" is accepted, according to the argument, the estimated number of black objects increases, while the estimated number of ravens does not change.

It can be illustrated by considering the situation of two people who have identical information regarding ravens and black objects, and who have identical estimates of the numbers of ravens and black objects. For concreteness, suppose that there are 100 objects overall, and, according to the information available to the people involved, each object is just as likely to be a non-raven as it is to be a raven, and just as likely to be black as it is to be non-black:

<math display="block">P(Ra)=\frac{1}{2} \ \ \ \ \ \ \ \ P(Ba)=\frac{1}{2}</math>

and the propositions <math>Ra,\ Rb</math> are independent for different objects <math>a</math>, <math>b</math> and so on. Then the estimated number of ravens is 50; the estimated number of black things is 50; the estimated number of black ravens is 25, and the estimated number of non-black ravens (counterexamples to the hypotheses) is 25.

One of the people performs a statistical test (e.g. a [[type I and type II errors|Neyman-Pearson]] test or the comparison of the accumulated weight of evidence to a threshold) of the hypothesis that "All ravens are black", while the other tests the hypothesis that "All non-black objects
are non-ravens". For simplicity, suppose that the evidence used for the test has nothing to do with the collection of 100 objects dealt with here. If the first person accepts the hypothesis that "All ravens are black" then, according to the argument, about 50 objects whose colors were previously in doubt (the ravens) are now thought to be black, while nothing different is thought about the remaining objects (the non-ravens). Consequently, he should estimate the number of black ravens at 50, the number of black non-ravens at 25 and the number of non-black non-ravens at 25. By specifying these changes, this argument ''explicitly'' restricts the domain of "All ravens are black" to ravens.

On the other hand, if the second person accepts the hypothesis that "All non-black objects are non-ravens", then the approximately 50 non-black objects about which it was uncertain whether each was a raven, will be thought to be non-ravens. At the same time, nothing different will be thought about the approximately 50 remaining objects (the black objects). Consequently, he should estimate the number of black ravens at 25, the number of black non-ravens at 25 and the number of non-black non-ravens at 50. According to this argument, since the two people disagree about their estimates after they have accepted the different hypotheses, accepting "All ravens are black" is not equivalent to accepting "All non-black things are non-ravens"; accepting the former means estimating more things to be black, while accepting the latter involves estimating more things to be non-ravens. Correspondingly, the argument goes, the former requires as evidence ravens that turn out to be black and the latter requires non-black things that turn out to be non-ravens.<ref name=O'Flanagan2008/>

====Existential presuppositions====
A number of authors have argued that propositions of the form "All <math>A</math> are <math>B</math>" presuppose that there are objects that are <math>A</math>.<ref>Strawson PF (1952) ''Introduction to Logical Theory'', Methuan & Co. London, John Wiley & Sons, New York</ref> This analysis has been applied to the raven paradox:<ref name=Cohen1987>{{cite journal |last=Cohen |first=Yael |date=March 1987 |title=Ravens and relevance |journal=[[Erkenntnis]] |volume=26 |issue=2 |pages=153–179 |doi=10.1007/BF00192194|s2cid=122284270 }}</ref>

{{quote|... <math>H_1</math>: "All ravens are black" and <math>H_2</math>: "All nonblack things are nonravens" are not ''strictly equivalent'' ... due to their different existential presuppositions. Moreover, although <math>H_1</math> and <math>H_2</math> describe the same regularity – the nonexistence of nonblack ravens – they have different logical forms. The two hypotheses have different senses and incorporate different procedures for testing the regularity they describe.}}

A modified logic can take account of existential presuppositions using the presuppositional operator, '*'. For example,

<math display="block">\forall\ x,\ *Rx\rightarrow Bx</math>

can denote "All ravens are black" while indicating that it is ravens and not non-black objects which are presupposed to exist in this example.

{{quote|... the [[logical form]] of each hypothesis distinguishes it with respect to its recommended type of supporting evidence: the possibly true [[substitution instance]]s of each hypothesis relate to different types of objects. The fact that the two hypotheses incorporate different kinds of testing procedures is expressed in the formal language by prefixing the operator '*' to a different predicate. The presuppositional operator thus serves as a relevance operator as well. It is prefixed to the predicate '<math>x</math> is a raven' in <math>H_1</math> because the objects relevant to the testing procedure incorporated in "All raven are black" include only ravens; it is prefixed to the predicate '<math>x</math> is nonblack', in <math>H_2</math>, because the objects relevant to the testing procedure incorporated in "All nonblack things are nonravens" include only nonblack things. ... Using [[Fregean]] terms: whenever their presuppositions hold, the two hypotheses have the same [[sense and reference|referent]] (truth-value), but different [[sense and reference|senses]]; that is, they express two different ways to determine that truth-value.<ref name=Cohen1987/>}}