Boole's Principles of Symbolical Reasoning

TL;DR

This paper examines Boole's Principles of Symbolical Reasoning, clarifies their historical context, and provides a corrected interpretation applicable to his algebra of logic for classes.

Contribution

It offers a detailed analysis of Boole's Principles, correcting and clarifying their application to his algebra of logic, and situates them within the historical development of algebra.

Findings

Boole's Principles are a generalization of Peacock's algebraic work.

A corrected version of Boole's Principles is proposed.

The principles are applicable to algebra of logic for classes.

Abstract

In modern algebra it is well-known that one cannot, in general, apply ordinary equational reasoning when dealing with partial algebras. However Boole did not know this, and he took the opposite to be a fundamental truth, which he called the Principles of Symbolical Reasoning in his 1854 book {\em Laws of Thought}. Although Boole made no mention of it, his Principles were clearly a generalization of the earlier publications on algebra by the Cambridge mathematician Peacock. After a detailed examination of Boole's presentation of his Principles, we give a correct version that is applicable to his algebra of logic for classes.