Carl Hempel’s “Paradox of the Ravens,”

proposed in 1965 through his “Studies in the Logic of Confirmation,” introduces

a problem where inductive reasoning disagrees with own intuition in response to

Nicod’s Criterion of Confirmation. Hempel states that a law where A calls for B

is validated if a recording of B corresponds with A, and that same law is

invalidated if A does not correspond with B. In other words, a hypothesis is

confirmed by observation and disproven where it is observed either to be

incorrect or incorrectly. Seemingly simple to understand, yet this idea comes

with several pieces of baggage, one being the Paradox of the Ravens.

Hempel introduces this paradox

through two hypotheses: A, whereas all ravens are black, and B, whereas all

non-black things are non-ravens. These two hypotheses are logical and equal, as

they are different versions of the same hypothesis, arguing the same content.

According to Nicod’s initial criterion, observing a black raven confirms A, all

ravens are black, but is neutral to B, all non-black things are non-ravens.

Very similar to as the observation of a non-black non-raven would confirm B but

be neutral to A, this violates the equivalence condition – as whatever confirms

or disconfirms one of two equivalent hypotheses also must confirm or disconfirm

the other. This brings the Paradox of the Ravens into light, as observation

supporting one hypothesis cannot be used to support another hypothesis that is

technically logically equivalent to the first. Different philosophers have

attempted to understand and solve this paradox, with the most successful being

the Bayesian confirmation theory being used hand-in-hand with Hempel’s original

criterion.

Quine and Foster try to solve the

paradox through the deemed “projectability” of a hypothesis. Projectable

hypothesis are confirmations generalized by their occasions according to Nicod’s

criterion of confirmation, more or less hypotheses made up of predicates

commonly used in projections. The Foster-Quine argument says although all

ravens are black is a projectable statement, all non-black things are

non-ravens is not projectable because “non-black” is a projectable predicate. All

non-black things being non-ravens is not projectable, and the observation of

non-black non-ravens is not confirming, so therefore the observation of a

non-black non-raven does not confirm B either. Likewise, the observation of a

black raven does actually confirm both A and B on account of the projectability

of A that all ravens are black. According to the Foster-Quine argument, the

equivalency condition should not be denied after the difference in projectability

is taken into account.

This account, however, is not conclusive, as the

paradox itself comes out of the idea that non-black non-ravens do not confirm

A, but the solution of the paradox claims the observations of black ravens is

used to support B. Therefore, the Foster-Quine argument takes away from this

idea’s persuasiveness, but can be explained through projectability. However,

the Foster-Quine argument creates another paradox in itself, stating that a

black raven can confirm that “all non-black things are non-ravens,” but a

non-black non-raven cannot do so. Goodman and Schleffer point out that the

Foster-Quine argument misinterprets the general idea of projectability, as it

is not the nature of the predicate that is the determining factor whether a

hypothesis is projectable or not, but it is always still projectable as long as

no conflicting hypotheses are present. Therefore, the Foster-Quine argument

does not show B to be projectable and this argument loses foundation.

Goodman and Schleffer argue that A

does logically differ from B, as A has an excluded contrary that all ravens are

not black while B has the excluded contrary that all non-black things are

ravens. As these two contraries conflict, the observation of a non-black

non-raven does not support A, that all ravens are black, as according to

Goodman and Schleffer, the two hypotheses are not logically equivalent. The

observation of a non-black non-raven does not support B as it also invalidates

the contrary. Also, an observation of a black raven supports A but does not

support B, as it satisfies its contrary that all non-black things are ravens,

in the way where it does not invalidate it. This introduces the idea of

selective confirmation, whereas that an observation that supports a conditional

hypothesis must also not invalidate its contrary. The observation of a

non-black non-raven does not confirm A, that all ravens are black, any more or

any less than it confirms that all ravens are not black.

Hempel himself tried to solve the

paradox by arguing that it wasn’t a paradox at all, and that it was 100%

logical for the observation of a non-black, non-raven to confirm in some way

that all ravens are black. Hempel argued that A, all ravens are black, suggest

that everything in the universe is either not a raven or is black, so observing

anything with both conditions satisfies is a confirmation to the paradox in

itself. With this being said, observing a non-black non-raven confirms A as it

satisfies these conditions, just as seeing a black raven or a yellow banana

does. This is known as Hempel’s Satisfaction Criterion of Confirmation, which

corresponds with the Bayesian Confirmation Theory.

The Bayesian Confirmation Theory is

based on mathematical probabilities, starting with the assumption that a person

can have different degrees of certainty about a belief. These degrees are

described with any number between 0 and 1, with values closer to 1 denoting a

certainty about a belief. The conditional probability of A given B is P(A/B) =

P(A and B)/P(B). If a piece 35to equal the probability of E, written P(H/E),

equaling P(H and E)/P(E). So, the Bayesian confirmation theory attempts to

solve the paradox of the ravens by arguing the observation of a non-black

non-raven does support the hypothesis that all ravens are black, to a

negligible extent.

So, according to the Bayesian

theory, observing a black raven will double the belief of the Hypothesis, P(H),

the conditional probability given for HG to double is that of the initial

probability. For the Bayesian Theory to work it must be possible for a single

observation to have an effect on each of the subjunctive probabilities. With

either the Foster-Quine theory of confirmation or the Selective Confirmation

Model the Bayesian Theorem would not hold.

I

do believe it is plausible to believe observing every non-black thing in the

world and discovering that none of them are ravens would have strong support

for the idea that all ravens are black, if one could observe that. Both pieces

of evidence account for many intuitions in terms of confirmation, like how each

successful confirming observation of seeing a black raven will cause P(H) to

increase by a lesser amount than the increase from a previous observation.

Bayesian theory does not suggest that it would be logical to base a hypothesis

such as all ravens are black on an observation of a non-black non-raven as such

an observation will cause the P(H) to increase by such a miniscule amount that

no difference is made. Only the observation of an actual black raven can

increase P(H) enough for a different hypothesis to be able to be entertained.

By reading into the Bayesian theorem, and Hempel’s initial satisfaction

criterion, the Paradox of the Ravens is able to be solved