Paradox of Confirmation Paradoxes seem to form the essence of irrationality and to continuously prove that rationality has a limit and that rationally inducing a fact may in fact prove the fact wrong. What is in fact a paradox? If we follow one definition, a paradox is "a parody on proof. It begins with realistic premises, but the conclusion falsifies these premises." More so, however, a paradox "arises when a set of apparently incontrovertible premises gives unacceptable or contradictory results." (Blackburn 1996, p. 276).

Before starting to analyze one of the most well-known paradoxes in philosophy, let us first have a look at three of the more important concepts of evidence, the classificatory concept, the comparative and the quantitative concept. Classificatory concepts are "those which serve for the classification of things or cases into two or a few mutually exclusive kinds." This concept helps divide a larger set into a series of smaller subsets with the obvious advantage that it is much easier to analyze the characteristics of a set with fewer elements. In our case, related to the ravens paradox, such a concept would be represented by classifying all objects into ravens or non-ravens, as well as into black and non-black objects.

The quantitative concepts serve to "characterize things or events or certain of their features by the ascription of numerical values." Indeed, several things, especially physical characteristics, can be described by using numerical value. Things like height or weight can easily have a number associated to them. In our case, we can relate for example the number of ravens in the world. Using this quantitative concept when discussing the paradox will help in the end make probabilistic assumptions about the statement "all ravens are black."

Comparative concepts, on the other hand, "serve for the formulation of the result of a comparison in the form of a more-less-statement without the use of numerical values." Seeing the discussion of our paradox, such a concept can be applied in finally stating that it is more or less probable that our statement is true given a certain fact.

Some preliminary approaches to paradoxes: Nicod's Criterion of Confirmation and The Equivalence Condition

Nicod's criterion of confirmation is one of the most important evidences around which the raven paradox will revolve. The criterion basically proposes the following statements:

For every x if x is a P. then it follows that x is a Q.

A a) A confirming instance would be, x is P. And x is Q (Px & Qx) b) A disconfirming instance would be, x is P. And x is not Q (Px & ~Qx).

A c) A neutral or irrelevant instance would be, x is not P (~Px).

Now to turn to the equivalence condition. Humberstone argues that such a condition is reasonable because "whether or not a hypothesis is confirmed by an observation should depend on the content of the hypothesis and not on the way that it happens to be formulated." Hence, "logically equivalent formulas have the same content." Other wise put, the equivalence condition states that if we have two hypotheses H. And H', logically equivalent and a proposition E. that confirms H, than it will confirm H'. Applied to the ravens paradox that will be discussed below, the observation of a purple cow will confirm statement H' that says that "all non-black objects are not ravens," hence it will also confirm its logical equivalent "all ravens are black."

Now, as we see from the lines above, the equivalence condition and Nicod's criterion lead to a paradox situation by themselves. If observations that confirm a hypothesis confirm anything logically equivalent, then this contradicts Nicod's statement that a non-As non-Bs are irrelevant.

The Paradox of the Ravens

Carl Hempel was the first to publish the paradox of the ravens in Theoria, a Swedish periodical, in 1937, and ever since, the paradox has been a source of numerous controversies. In his paper, Hempel concludes that the generalization of a simple statement, such as "all ravens are black" can be confirmed by another simple observation, such as that of a purple cow. The observation of a purple cow would, in Hempel's opinion, increase, even slightly, the probability that all ravens are black. Briefly summarizing his paradox, professor Hempel notes that the statement "all ravens are black" is logically equivalent to the statement that "all non-black objects are not ravens" (this is a true logical equivalence). Therefore, finding a purple cow weakly confirms the statement that all ravens are black, because...

As Hempel states, observing a non-black object will confirm his statement that all ravens are black to "an infinitesimal degree." The more observations of non-black objects, the stronger the statement will be confirmed. The catch of Hempel's paradox would be that the observation of a purple cow also confirms the statement that "all ravens are white," following the same logical discussion ("all non-white objects are not ravens, hence a purple cow will confirm this statement and its logical equivalent that all ravens are white). How can a purple cow then confirm two opposite statements that "all ravens are white" and "all ravens are black"?
This was but a brief description of the paradox. Let us now dig in further and discover some probabilistic and logical assumptions about this statement. The paradox itself is constituted of three propositions:

1) Observations of black ravens confirm 'All ravens are black'.

2) Observations of purple cows, white swans, etc., are neutral to (i.e. do not confirm) 'All ravens are black'.

3) If observations confirm one formulation of a hypothesis they confirm any logically equivalent formulation

The three propositions are themselves incompatible with one another. For example, an observation of a purple cow will confirm "all non-black objects are not ravens" (A), hence also "all ravens are black" (B) (deductible from proposition number 3: observation that confirms A true implies B. true). However, the same observation denies the neutrality derived from 2. It can thus be concluded that proposition 3 cannot be denied (it is a logically proven fact), hence, the paradox could be solved by denying either 1 or 2. As we have seen in the brief presentation of the paradox, professor Hempel denies proposition 2 and concludes that any observation of non- black objects will infinitesimally prove the statement.

Intuitively, we can state that proposition 2 is more likely to be rejected than proposition 2. This is because the most obvious proof of the statement "all ravens are black" would be observations of black ravens. Hence, if we are to reject proposition 2, then Hempel's solution to the paradox is right in this context and Nicod's definition can be rejected as well.

It is time to introduce a further dilemmatic element: we can add the information that there are far more non-black things than ravens. If we regard this additional information, we can still deny proposition 2, regarding neutrality, but we can now affirm that observing non-black non-ravens, such as a purple cow, is not as good a proof of the statement as observing black ravens. This does not reject Hempel's assumption (that rejects proposition number 2), but somewhat limits the context in which it is placed.

We may assume that from the three propositions mentioned above, we can accept proposition 3 as a logical axiom, as well as proposition 1, and reject proposition 2. However, it is notable to make a brief reference to Professor Watkin's view, in which he accepts both propositions 1 and 2. The statement we are dealing with includes two elements: ravens and black. Let us assume one of these fixed (that is, we will be assuming that we are observing an object, which we KNOW is black), hence, the outcome of the observation can either be that it is a raven or that it is not a raven. Thus, the observation will either confirm the hypothesis or it will not (in the sense that if the outcome is the other way, it does not confirm). The fact that two different observations cannot both confirm our hypothesis means that if we observe an object that we know is a raven and it turns out to be black, this confirms out hypothesis, but if we inspect a black object and it turns put to be a raven, this does not.

We have entered an area of ambiguous hypothesis and conclusions, that is why I would like to restate the paradox and follow up a different approach to it. However, the observations made here above still stand as assumptions and discussions around the paradox.

The paradox itself seems to revolve around two elements: the first that the observation of a purple cow really has nothing to do with the generalization of the statement "all ravens are black" and the second that such an observation also proves that "all ravens are white." Let us have a look at the first paradox. Solving it would either mean contradicting Hempel in his conclusion that the observation of a purple cow actually proves (even if infinitesimally) that all ravens are black or that the common…