The reference class problem in Hempel’s paradox

(adapted from Franceschi (1999) & (2014))

Hempel’s paradox is based on the fact that the two following assertions:

(H) All ravens are black

(H*) All non-black things are non-ravens

are logically equivalent. By its structure (H*) presents itself indeed as the contrapositive form of (H). It follows that the discovery of a black raven confirms (H) and also (H*), but also that the discovery of a non-black thing that is not a raven such as a red flame or even a grey umbrella, confirms (H*) and therefore (H). However, this latter conclusion turns out to be paradoxical.

We shall endeavour now to detail the dichotomous analysis on which is based the solution proposed in Franceschi (1999). The corresponding approach is based on finding a reference class associated with the statement of the paradox, which may be defined with the help of an A/Ā duality. If we scrutinise the concepts and categories that underlie propositions (H) and (H*), we first note that there are four categories:

  • ravens
  • black objects
  • non-black objects
  • non-black objects.

It turns out that three of the four classes do not pose any particular problem. To begin with, a raven is precisely defined within the taxonomy in which it inserts itself. A category such as that of the ravens can be considered well-defined, since it is based on a precise set of criteria defining the species corvus corax and allowing the identification of its instances. Similarly, the class of black objects can be accurately described, from a taxonomy of colours determined with respect to the wave lengths of light. Finally, we can see that the class of non-black objects can also be a definition that does not suffer from ambiguity, in particular from the specific taxonomy of colours which has been just mentioned.

A blue blackbird

However, what about the class of non-ravens? What does constitute then an instance of a non-raven? Intuitively, a blue blackbird, a red flamingo, a grey umbrella and even a natural number, are non-ravens. But should we consider a reference class that goes up to include abstract objects? Should we thus consider a notion of non-raven that includes abstract entities such as integers and complex numbers? Or should we limit ourselves to a reference class that only embraces the animals? Or should we consider a reference class that encompasses all living beings, or even all concrete things, also including this time the artefacts? Finally, it follows that the initial proposition (H*) is susceptible of giving rise to several variations, which are the following:

(H1*) All that is non-black among the corvids is a non-raven

(H2*) All that is non-black among the birds is a non-raven

(H3*) All that is non-black among the animals is a non-raven

(H4*) All that is non-black among the living beings is a non-raven

(H5*) All that is non-black among the concrete things is a non-raven

(H6*) All that is non-black among the concrete and abstract objects is a non-raven

Thus, it turns out that the statement of Hempel’s paradox and in particular of proposition (H*) is associated with a reference class, which allow to define the non-ravens. Such a reference class can be assimilated to corvids, birds, animals, living beings, concrete things, or to concrete and abstract things, etc.. However, in the statement of Hempel’s paradox, there is no objective criterion for making such a choice. At this point, it turns out that one can choose such a reference class restrictively, by assimilating it for example to corvids. But in an equally legitimate manner, we can choose a reference class more extensively, by identifying it for example to the set of concrete things, thus notably including umbrellas. Why then choose such or such reference class defined in a restrictive way rather than another one extensively defined? Indeed, we are lacking a criterion allowing to justify the choice of the reference class, whether we proceed by restriction or by extension. Therefore, it turns out that the latter can only be defined arbitrarily. But the choice of such a reference class proves crucial because depending on whether you choose such or such class reference, a given object such as a grey umbrella will confirm or not (H*) and therefore (H). Hence, if we choose the reference class by extension, thus including all concrete objects, a grey umbrella will confirm (H). On the other hand, if we choose such a reference class by restriction, by assimilating it only to corvids, a grey umbrella will not confirm (H). Such a difference proves to be essential. In effect, if we choose a definition by extension of the reference class, the paradoxical effect inherent to Hempel’s paradox ensues. By contrast, if we choose a reference class restrictively defined, the paradoxical effect vanishes.

The foregoing permits to describe accurately the elements of the preceding analysis of Hempel’s paradox in terms of one-sidedness bias such as it has been defined above: to the paradox and in particular to proposition (H*) are associated the reference class of non-ravens, which itself is susceptible of being defined with regard to the extension/restriction duality. However, for a given object such as a grey umbrella, the definition of the reference class by extension leads to a paradoxical effect, whereas the choice of the latter by restriction does not lead to such an effect.

Franceschi, P., « Comment l’urne de Carter et Leslie se déverse dans celle de Carter », Canadian Journal of Philosophy, vol. 29, Mars 1999, pages 139-156, The Doomsday Argument and Hempel’s problem (English translation)

Franceschi, P., “Éléments d’un contextualisme dialectique” (in english), in Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (éd.), Université de Genève, 2014, p. 581-608.

Ambiguous images Arbitrary focus Bistable perception Complementarity relationship Conflict resolution Conflict resolution with matrices of concepts Conflict types relating to matrices of concepts Contrary relationship Courage Dialectical contextualism Dialectical monism Dialectical monism in Aztec philosophy Dialectical monism in Heraclitus Dichotomic analysis Dichotomic analysis applied to paradox resolution Dichotomous reasoning Disqualification of one pole Disqualification of the positive Doctrine of the mean Doomsday argument Dualities Dual poles Extreme opposition General cognitive distortions Instance of one-sidedness bias Liar paradox Matrix of concepts Maximization Mental filter Minimization Bistable cognition Omission of the neutral One-sidedness bias One-sided viewpoint Opposition relationship Principle of dialectical indifference Requalification into the other pole Reference class Reference class problem Reference class problem in philosophical paradoxes Reference class problem in the Doomsday argument Reference class problem in Hempel’s paradox Reference class problem in the surprise examination paradox Selective abstraction Sorites paradox Specific cognitive distortions Surprise examination paradox System of taxa Two-sided viewpoint Viewpoint of a duality Viewpoint of a pole