On the Peirce's "balancing reasons rule" failure in his "large bag of beans" example

TL;DR

This paper examines a logical inconsistency in Peirce's 'balancing reasons rule' using a probability problem involving beans, highlighting why the rule fails in certain Bayesian updating scenarios.

Contribution

The paper analyzes Peirce's rule failure through a detailed example, clarifying the limitations of balancing reasons in probabilistic reasoning.

Findings

Twenty consecutive black observations increase confidence in black outcome.

Observation of nearly equal black and white counts leads to indifference.

Peirce's rule does not hold in Bayesian updating with accumulated evidence.

Abstract

Take a large bag of black and white beans, with all possible proportions considered initially equally likely, and imagine to make random extractions with reintroduction. Twenty consecutive observations of black make us highly confident that the next bean will be black too. On the contrary, the observation of 1010 black beans and 990 white ones leads us to judge the two possible outcomes about equally probable. According to C.S. Peirce this reasoning violates what he called "rule of balancing reasons", because the difference of "arguments" in favor and against the outcome of black is 20 in both cases. Why? (I.e. why does that rule not apply here?)