How did George Boole turn logic into algebra?

For more than two thousand years, logic, the study of correct reasoning, was taught mostly in words, in the style of the ancient Greek philosopher Aristotle. George Boole, a largely self-taught English mathematician working in Ireland, believed reasoning could be written down and calculated like algebra. In 1854 he set out that idea in full in a book with a long title: "An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities."

Most people call it simply The Laws of Thought. Its key move, letting symbols stand for classes of things and allowing them only the values 0 and 1, is why the word "Boolean" now appears in every programming textbook.

Who Boole was

Boole was born on 2 November 1815 in Lincoln, England, the son of a shoemaker whose real passion was science. The family had little money, and Boole had no elite schooling. With access to good books, he taught himself languages and advanced mathematics. From the age of sixteen he worked as a teacher, and at nineteen he opened his own small school in Lincoln, which he ran for about fifteen years while doing serious mathematical research on the side.

In 1847 he published a first, shorter book on the subject, The Mathematical Analysis of Logic. In 1849 he moved to Ireland as the first person to hold the chair of mathematics at Queen's College in Cork, which had just opened. He wrote the preface of The Laws of Thought there, dating it 30 November 1853, and the book came out in 1854. He regarded it as the mature statement of his ideas. Boole died in Cork on 8 December 1864, after a chill turned into pneumonia.

What the book set out to do

Boole states his aim on the first page: to find the basic laws of the mental operations we use when we reason, to write those laws in the symbols of a calculus, and to build logic and probability on that foundation. He even hoped the results would hint at how the human mind is constituted.

The method is surprisingly simple to state. Let a letter such as x stand for a class of things, say "white things," and y for another class, say "sheep." Writing them side by side, xy, means things that belong to both classes: white sheep. Boole then noticed something odd. If x and y mean the same class, then xy is just x again. Selecting "white things" from "white things" changes nothing. So in his logic:

x × x = x, which he wrote as x*² = *x

In ordinary arithmetic that equation is true for only two numbers: 0 and 1. Boole concluded that the algebra of logic is the same as ordinary algebra with the values limited to 0 and 1. He gave those values a meaning: 1 stands for the whole "universe" of things under discussion, and 0 stands for nothing, the empty class. Rearranging his law gives x(1 − x) = 0, which says that nothing can both belong and not belong to a class. Boole pointed out that this is Aristotle's principle of contradiction, now derived by algebra.

A worked example: logic as arithmetic

Treat each statement as 1 if true and 0 if false. Boole's multiplication then behaves like the word AND, and 1 − x behaves like NOT.

Check the law: if x = 1, then 1 × (1 − 1) = 0; if x = 0, then 0 × (1 − 0) = 0. Either way the result is 0, so the principle of contradiction holds for every case. A whole argument can now be checked by working through every combination of 0s and 1s, a job that requires no insight, only patience. That is exactly the kind of job a machine is good at.

Why it matters

The book's reception disappointed Boole a little; readers treated it mainly as a mathematical toolkit rather than a study of the mind. Others built on it. The English economist William Stanley Jevons simplified his system in 1864, and the American philosopher Charles Sanders Peirce gave the name "Boolian algebra" to the streamlined version. With a change of spelling, "Boolean algebra" spread through universities in the early twentieth century.

Boole's algebra was not exactly modern Boolean algebra. He kept closer ties to ordinary arithmetic, and scholars point out the differences. But the central idea, that reasoning with true and false can be done by calculation on 0 and 1, comes from him. In 2019 you meet it every time you type AND or OR into a library search, write an "if" statement in code, or rely on any digital device.

How does Boole's logic end up inside a computer?

For decades Boole's logic looked like pure theory with no engineering use. That changed in 1937, when a young student named Claude Shannon wrote his master's thesis at MIT, the Massachusetts Institute of Technology. Shannon had met Boole's ideas in a philosophy class, and he saw that an electrical switch is a perfect 0 or 1: open or closed. He showed that Boolean algebra could be used to design and simplify circuits built from relays, the switches then used in telephone exchanges, and that such circuits could in turn solve problems in Boolean algebra. In the Soviet Union, Victor Shestakov reached a similar theory of switching circuits around the same time.

Modern chips use the same principle at enormous scale. Tiny transistors act as switches, grouped into gates that compute AND, OR and NOT, and billions of them together run every program. Even the neural networks behind the speech recognition and image tools of 2019, which learn from data rather than following hand-written rules, run on hardware built entirely from Boole's two values. Shannon, in turn, was one of the four authors of the 1955 proposal that gave artificial intelligence its name, so the line from Boole's book to AI research is a short one.