The Raven Paradox: the Case of Infinite Sets and the Case of Finite Sets

Infinite sets and finite sets have different properties and follow different rules. C “All the elements of the set of ravens are elements of the set of black ravens”. CE “All the things that are not elements of the set of black ravens are elements of the set of the things that are not ravens”. It is assumed that the number of ravens is infinite, but it is not assumed that the number of black ravens is infinite: that the number of black ravens is infinite is only a hypothesis (for which confirmation is sought), it cannot be assumed as true that the number of black ravens is infinite (the set of ravens could contain an infinite number of red ravens and only a finite number of black ravens). Therefore, in the generalization C we refer to a set that is assumed to be an infinite set (the set of ravens) and to a set (the set of black ravens) that is not assumed to be an infinite set. By contrast, in the generalization CE we refer to two sets that are assumed to be infinite sets (the set of all objects that are not elements of the set of black ravens and the set of all objects that are not elements of the set of ravens). That CE has infinite instances is something we already know a priori (by virtue of the assumptions made); by contrast, we do not know a priori that C has infinite instances—that is merely an aspiration. Therefore, an instance of C carries weight, whereas an instance of CE tells us nothing more than what we already know a priori. To summarize the case of finite sets: ​1) The Disparity in Set Cardinality The set of ravens is vastly smaller than the set of non-black things that are non-ravens. In any plausible finite universe, the number of non-black objects is overwhelmingly larger than the number of ravens. Therefore, while investigating the restricted set of ravens significantly reduces the number of potential counterexamples, sampling from the near-infinite reservoir of non-black objects leaves the hypothesis virtually untested. ​2) The Problem of Indiscriminate Confirmation The observation of a green emerald is a positive instance of the contrapositive hypothesis CE ("All non-black things are non-ravens"). However, by the exact same logical token, that very same green emerald is also a positive instance of DE ("All non-yellow things are non-ravens", which is logically equivalentto D "All ravens are yellow"). If we accept that observing a non-black non-raven confirms that all ravens are black, we must also accept that it confirms that all ravens are yellow. Because a single piece of external evidence would simultaneously confirm two mutually exclusive hypotheses, looking outside the target set lacks any real discriminative power. ​3) Parallelism with the Infinite Case: Information vs. Redundancy There is a parallelism with the infinite sets scenario based on our background knowledge. That the universe contains an immense, virtually inexhaustible number of non-black things that are non-ravens is a structural fact already fully possessed by our background knowledge; it is already baked into our prior assumptions about the world. Consequently, observing a non-black thing that is a non-raven (for example, a green emerald) adds zero new information—it merely repeats an a priori platitude. Observing a black raven adds new information.