In this essay, I will first describe the origin and nature
of the Paradox of the Ravens. Then I
will briefly describe some of the less popular responses to the puzzle, before
continuing with a description of the response most widely accepted in the
literature. I will, however,
side-step the issue of how to define "best" in this context.
Instead, I will conclude by documenting why I agree with the general
opinion that the Bayesian analysis is sufficiently useful to be regarded as
"the" response to the paradox of the ravens.

The "Paradox of the Ravens" was proposed by the German
logician Carl Gustav Hempel in the 1940s to illustrate a problem where inductive
logic seems to violate intuition^{(1)}.
Hempel proposed the inductive hypothesis "All ravens are black", and
explored how it is that we confirm this hypothesis.
In strict logical terms, by the Contrapositive Law of deductive logic,
this hypothesis is equivalent to "Everything that is not black is not a raven".
Clearly, the observation of a white shoe is evidence in support of this
latter hypothesis -- a white shoe is not black and is not a raven.
But since the two hypotheses are logically equivalent, the observation of
a white shoe must also be accepted as evidence in support of the hypothesis that
"All ravens are black". Which
appears to be counter-intuitive.

The apparent paradox relies on three premises that are
regarded as intuitively plausible, yet are logically inconsistent.

**Premise 1
(Nicod's Condition)** -- In the absence of other evidence, the observation that
some object **
a** is both

**Premise 2 (Equivalence Condition)** -- If evidence confirms a
proposition, then it also confirms any logically equivalent proposition.
The rules of deductive logic demand that the truth conditions of
logically equivalent propositions are the same.

**Premise 3** -- In the absence of other evidence, a white shoe does not
confirm the hypothesis that all ravens are black.

To "resolve" the paradox, obviously one of these three
premises must be denied. And to
provide a "good" resolution for the paradox (again, without getting into what
would define "good" in this context), that denial must rest on some solid
rationale.

I.J. Good^{(2)} and Patrick Maher^{(3)} are
two philosophers who have explored the potential of denying Nicod's Condition.
They each have pointed out that in certain conditions of background
knowledge, Nicod's Condition turns out to be false.
Consider, for example, someone who is familiar only with birds in the
parrot family. Parrots come in a
delightful multitude of colours.
When advised that a raven is a bird, such a person might reasonably conclude
that ravens, like other birds, would come in a multitude of colours.
Seeing a black raven would then not confirm the hypothesis that "all
ravens are black", but would instead confirm the hypothesis that "all ravens are
multi-coloured". Drawing on the
equivalency condition, this would entail that observing a white shoe would also
not confirm the hypothesis that all ravens are black.
So there is a reasonable case to be made that Nicod's Condition ought to
be dropped from the paradoxical argument.

Hempel was aware of this argument, but responded that the
paradox must be understood in the context of either no background information,
or actual rather than fanciful background information^{(4)}.
Dropping Nicod's Condition might be justifiable under some imaginable
conditions, but is clearly not justifiable given our actual background knowledge
of ravens in particular, and birds in general (i.e. that birds in the same
species are usually similarly coloured).
Similarly, it is clearly not justifiable if all background information is
to be rejected when considering the logic of the paradox.

W.V.O. Quine offered another approach to justify dropping
Nicod's Condition. Quine argued that
only certain "natural kind" predicates obey Nicod's Condition, while
artificially contrived predicates do not^{(5)}.
Hence, Nicod's Condition holds for black ravens because black ravens form
a "natural kind". But Nicod's
Condition does not hold for the artificial predicate "non-black non-ravens"
because it does not delimit a "natural kind".
This approach was offered in the context of Nelson Goodman's example of
the predicate "grue". But it serves
equally well as a resolution to the paradox of the ravens.
Hempel's ravens argument appears paradoxical, on this basis, because we
automatically apply Nicod's Condition to all predicates, when in fact it is only
valid when applied to "natural kinds".

Of course, Quine's response to the paradox only works if
one accepts the "natural kinds" approach to linguistic predicates.
There are many objections to the concept of "natural kinds", but I will
side-step that debate in this essay by simply noting that as a resolution of the
ravens paradox, the "natural kinds" justification for dropping Nicod's Condition
is not widely accepted.

Israel Scheffler and Nelson Goodman offered an argument for
dropping the Equivalence Condition premise^{(6)}.
Their notion of "Selective Confirmation" draws upon the theory of Karl
Popper^{(7)} that scientific hypotheses are never confirmed, only
falsified. The concept of "selective
confirmation" of a proposition is understood as a falsification of the contrary
proposition. The observation of a
black raven thus does not confirm the hypothesis that "all ravens are black".
Rather, it falsifies the contrary hypothesis that "no ravens are black".
In other words, selective confirmation violates the equivalency condition
because observing a black raven selectively confirms "all ravens are black" (by
falsifying its contrary), but not "all non-black things are non-ravens" (because
it does not falsify its contrary). A
white shoe (a non-black-non-raven) therefore does not selectively confirm the
hypothesis that "all ravens are black" because it is consistent with both "all
ravens are black" and "no ravens are black".

The notion of selective confirmation is one example of an
approach to the concept of "supporting evidence" that does not coincide with the
concept of "increased likelihood".
Since by the laws of the probability calculus, logically equivalent propositions
must have the same probabilities, any suggested resolution to the ravens paradox
that denies the equivalency condition must interpret "supporting evidence" in
terms that do not coincide with a notion of probability.
There are a number of approaches of this sort that have been offered in
the vast literature on the ravens paradox, but none has attracted wide support.
In the interests of brevity, therefore, I will not pursue them further.

The solution to the paradox that has achieved the widest
acceptance, is the denial of the 3^{rd} premise.
This means accepting the apparently paradoxical conclusion that the
observation of a white shoe (a non-black non-raven) does indeed support the
hypothesis that all ravens are black.

The most widely accepted argument in support of this
response to the ravens paradox, is the Bayesian analysis.
However, as Chihara^{(8)} has pointed out, there is not one
single Bayesian analysis. Rather
there is a multitude of slightly different analyses.
All of which take an anti-Popperian view of hypothesis confirmation, and
all of which adopt a "probability raising" notion of "supporting evidence".
Despite the range of alternative analyses of the scenario, there is a
common conclusion. Because of the
relative (subjectively estimated or historically observed) infrequency of ravens
to non-ravens and black things to non-black things, the observation of a white
shoe does increase the probability of the hypothesis that "all ravens are black"
but only to a very miniscule degree.

Formally, in one version, Bayes' Theorem states --

P(H/e.k) = __P(e/H.k) x P(H/k)__

P(e/k)

In words -- the (posterior) probability of the hypothesis
**H**, given the evidence
**e** and the background knowledge
**k** is equal to the probability of the
evidence **e** given the hypothesis
**H** and the background knowledge
**k**, times the (antecedent) probability
of the hypothesis **H** given the
background **k**, all divided by the
probability of the evidence **e** given
the background **k**.

In the Ravens scenario where the hypothesis
**H** is "All ravens are black", and
**e** is "a white shoe", the P(e/H.k) is
almost equal to P(e/k), so P(H/e.k) is almost equal to P(H/k).
P(e/k) is only minimally less than P(e/H.k) because the hypothesis that
"all ravens are black" minimally reduces the vast number of things that are
non-black and non-raven, when compared to the situation without that hypothesis.
The probability of seeing a white shoe, given all that we know about the
world, is only minimally less than the probability of seeing a white shoe, given
all that we know about the world combined with the hypothesis that all of one
collection of things (the ravens) will be black.

Premise 3 above gains its appearance of plausibility
because the degree to which a white shoe confirms that "all ravens are black" is
so very close to zero that for all practical purpose it can be (and hence is)
treated as zero.

Despite the wide acceptance of the Bayesian response to the
Ravens paradox, the Bayesian approach does suffer from a couple of difficulties.
One is major and serious, and one is minor and only sometimes serious.
The major and serious difficulty with any Bayesian analysis is that it
depends upon probabilities -- and we do not have a very good understanding of
just what constitutes a probability (or likelihood, or tendency).
In situations where we can approximate the probabilities involved using
the history of previously observed frequencies, this may not pose much of a
road-block. But in cases where we
have to provide some estimation of the antecedent likelihood of some hypothesis,
given only our current background information, this becomes a serious
difficulty. The philosophical
discussion of the meaning of such subjectively estimated probabilities is
voluminous. I do not intend to get into that morass here, except to note this
issue remains a problem for the Bayesian response to the ravens paradox.

The minor and only sometimes serious difficulty is that any
Bayesian analysis presupposes that the probability of the observation (given
background information) is independent of the hypothesis (or theory) being
examined^{(9)}. In other
words, the Bayesian analysis presupposes that P(e|H.k) ~= P(e.k).
In a case like the ravens hypothesis we can be reasonably sure from
background information that the presupposition holds good.
But this is not always the case.
The presupposition may be particularly problematic, for example, in
quantum physics -- where it is recognized that any observation of the evidence
alters the situation. To apply a
Bayesian analysis to any hypothesis in general, therefore, demands that we first
gain some reasonable assurance that the presupposition holds true and that the
formulation of the hypothesis does not impact the probability of the evidence.
This issue gains in importance when one considers the issue of the
theory-ladenness of observation, and the extent to which the theory we adopt can
change how we view the world, and the probabilities we attach to observing the
evidence we observe. But that will
remain the topic of another essay.

One of the lessons that we can learn from the Bayesian
response to the paradox of the ravens, is that whether or not observations
confirm hypotheses is never independent of background information.
Probabilities are the core of any Bayesian analysis of scenarios like
that of the white shoe and the ravens hypothesis.
And probabilities, whether conceived as subjective estimations, or as
observed historical frequencies, are "all things considered" evaluations.
As for that matter, is the assurance that the necessary presupposition of
hypothesis independence holds good.

I choose the Bayesian analysis of the Paradox of the Ravens
as the "best" response because it does not attack the intuitively obvious
premise that observing a black raven confirms (admittedly to some small degree)
the hypothesis that all ravens are black (Nicod's Condition).
Without that condition, we could not do the kind of inductive pattern
recognition that we are so good at.
And because it does not challenge the defined rules of Deductive Logic and the
Probability Calculus that logically equivalent propositions have equivalent
truth conditions (the Equivalency Condition).
And because the Bayesian analysis provides a ready explanation of why the
Premise 3 appears so intuitively reasonable while remaining strictly false.
Despite the challenge presented by a lack of a complete philosophical
understanding of the probabilities involved, and the potential challenge
presented by the theory-ladenness of observation, the Bayesian approach to
related problems of experimental confirmation of scientific hypotheses has
proved remarkably (albeit pragmatically) resourceful.

(1) Wikipedia
contributors. "Raven paradox" in Wikipedia, The Free Encyclopedia.
URL=<http://en.wikipedia.org/w/index.php?title=Raven_paradox&oldid=455568165>.

(2) Good, I.J.;
"The White Shoe is a Red Herring" in The British Journal for the Philosophy of
Science, Vol. 17, No. 4 (Feb 1967), p322.

(3) Maher,
Patrick; "Inductive Logic and the Ravens Paradox" in Philosophy of Science, Vol
66, No 1 (Mar 1999), pp 50-70.

(4) Hempel,
C.G.; "The White Shoe - No Red Herring" in The British Journal for the
Philosophy of Science, Vol. 18, No. 3 (1967), p. 239

(5) Quine,
W.V.O.; "Natural Kinds" in Ontological Relativity and other Essays. Columbia
University Press, New York, New York, 1969. p114

(6) Scheffler,
Israel & Goodman, Nelson; "Selective Confirmation and the Ravens: A Reply to
Foster" in The Journal of Philosophy, Vol. 69, No. 3 (Feb. 10, 1972), pp. 78-83

(7) Popper,
Karl; The Logic of Scientific
Discovery, RC Series Bundle, Routledge Classics, Routledge, New York New York,
2002 (1934), ISBN 0-415-27844-9.

(8) Chihara,
Charles, S.; "Some Problems for Bayesian Confirmation Theory" in The British
Journal for the Philosophy of Science, Vol. 38, No. 4 (Dec 1987), pp 551-560.

(9) Vranas,
Peter B.M.; "Hempel's Raven Paradox: A Lacuna in the Standard Bayesian Solution"
in The British Journal for the Philosophy of Science, Vol 55, No 3 (2004), pp
545-560.