On Nicod's Condition, Rules of Induction and the Raven Paradox

TL;DR

This paper investigates Nicod's Condition in the raven paradox, analyzing how background knowledge affects its validity and revealing a conflict with inductive reasoning principles.

Contribution

It provides a detailed analysis of Nicod's Condition under different background knowledge scenarios, highlighting its limitations and relation to inductive inference rules.

Findings

Nicod's Condition can be violated under certain background knowledge.

The violation occurs specifically in the case of known counts of objects.

The analysis is measure-independent, strengthening the generality of results.

Abstract

Philosophers writing about the ravens paradox often note that Nicod's Condition (NC) holds given some set of background information, and fails to hold against others, but rarely go any further. That is, it is usually not explored which background information makes NC true or false. The present paper aims to fill this gap. For us, "(objective) background knowledge" is restricted to information that can be expressed as probability events. Any other configuration is regarded as being subjective and a property of the a priori probability distribution. We study NC in two specific settings. In the first case, a complete description of some individuals is known, e.g. one knows of each of a group of individuals whether they are black and whether they are ravens. In the second case, the number of individuals having a particular property is given, e.g. one knows how many ravens or how many black things there are (in the relevant population). While some of the most famous answers to the paradox are measure-dependent, our discussion is not restricted to any particular probability measure. Our most interesting result is that in the second setting, NC violates a simple kind of inductive inference (namely projectability). Since relative to NC, this latter rule is more closely related to, and more directly justified by our intuitive notion of inductive reasoning, this tension makes a case against the plausibility of NC. In the end, we suggest that the informal representation of NC may seem to be intuitively plausible because it can easily be mistaken for reasoning by analogy.