An Investigation of the Laws of Thought
Logic is a kind of algebra — reasoning reduced to symbols that obey laws.
George Boole had a strange and powerful idea: that the rules of logic could be written down and calculated, like sums in algebra.
The idea, unpacked
We usually treat logic — “all A are B,” “this and that,” “either/or” — as a matter of words and careful thinking. Boole saw that it could be made into mathematics. Let a letter stand for a category (say x = “white things”), and let combining categories follow rules that look just like algebra: “both x and y” becomes a kind of multiplication, x times y; “not x” becomes 1 minus x; the whole universe of things is 1, and nothing at all is 0.
The trick that makes it work is that his letters only ever take two values: 0 or 1 — off or on, false or true. Picking the white things out of the white things just gives you the white things again, so x times x equals x. That one humble fact — impossible in ordinary arithmetic except for 0 and 1 — is the seed of a complete, exact logic you can actually compute with.
Where it came from
Boole was a largely self-taught Englishman, a shoemaker's son who became professor of mathematics at Queen's College in Cork, Ireland, without ever holding a university degree of his own. He had sketched the idea in a short pamphlet in 1847; in 1854 he set it out in full in this book, whose grand title — An Investigation of the Laws of Thought — announced his belief that he had found the mathematics behind reasoning itself. His friend Augustus De Morgan was chasing related ideas at the very same time. Boole died young, in 1864 at 49, never seeing what his algebra would become.
Why it mattered
For two thousand years, logic had been Aristotle's — careful but verbal, and hard to extend. Boole turned it into something you could write in symbols and solve by calculation, opening logic to the full power of mathematics. That move — treating reasoning as computation — is one of the deepest in the history of thought, and it set the stage for everything from mathematical logic to the design of computers.
Light switches
Picture two light switches wired to one lamp. Call each switch 0 when off and 1 when on. Wire them in a row, one after the other, and the lamp lights only when both are on — that is Boole's “and,” x times y. Wire them side by side and the lamp lights when either is on — that is his “or.” A switch that flips the signal is “not.” Boole worked this algebra out in 1854 with no machine in mind; a century later it turned out to describe exactly how the circuits inside a computer think. Flip the switches below and watch the logic happen.
Where it sits
Boole stands at a hinge in the Library's story of reasoning. Behind him is Euclid, who showed that truths can be derived by pure logical steps; ahead of him are Gödel, who probed the limits of formal systems, and Claude Shannon, whose work carried Boole's two-valued algebra straight into the digital age. When you run a search with AND and OR, or when any computer adds two numbers, you are using the algebra Boole laid down in this book.
The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of Probabilities; and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind.
Let us conceive, then, of an Algebra in which the symbols x, y, z, &c. admit indifferently of the values 0 and 1, and of these values alone.
That axiom of Metaphysicians which is termed the principle of contradiction, and which affirms that it is impossible for anything to possess a quality, and in the same time not to possess it, is a consequence of the fundamental law of thought, whose expression is x² = x.