The Laws of Boole’s Thought

The algebra of logic developed by Boole was not Boolean algebra. In this article we give a natural framework that allows one to easily reconstruct his algebra and see the difficulties it created for his successors. It is well known that modern Boolean algebra is connected with the work of George Boole [1815–1864], namely with his two books on a mathematical treatment of logic: • The Mathematical Analysis of Logic, 1847 • The Laws of Thought, 1854. These two books will be referred to as Analysis and Laws. What is not well known is just how far removed Boole’s work was from modern Boolean algebra, both in substance and in spirit. Contrary to popular belief Boole did not work with a two-element Boolean algebra, nor with the Boolean algebra of subsets of a given set. Boole was simply not doing Boolean algebra, nor Boolean rings. As a matter of fact, for more than a century no one really knew why Boole’s algebra of logic worked. In 1869 Jevons said ([6], §7): The quasimathematical methods of Dr. Boole especially are so magical and abstruse, that they appear to pass beyond the comprehension and criticism of most other writers, and are calmly ignored. Jevons was one of the rare writers on logic to admit that Boole’s system was not built on principles that he could understand. But even Jevons, like so many others, had come to the conclusion that indeed Boole’s system worked ([7], §175): It is not to be denied that Boole’s system is consistent and perfect within itself. It is, perhaps, one of the most marvellous and admirable pieces of reasoning ever put together. Indeed, if . . . the chief excellence of a system is in being reasoned and consistent within itself, then Professor Boole’s is nearly or quite the most perfect system ever struck out by a single writer. Boole wanted a symbolic algebra for sets A, B, etc. He was fully aware of our favorite concepts of union, intersection, and complement (as well as symmetric difference). But rather than adopting our modern approach of assigning symbols to these operations and then determining the laws that they satisfy, that is, the idempotent, commutative, etc., laws, Boole approached the task from a Actually Boole followed the logicians of his time and used the word class. But for a modern audience the word set seems preferable. See Laws, p. 56, to find Boole’s mode of expressing the union and symmetric difference.