Logic IS algebra. AND=×, OR=+, NOT=1−x, true=1, false=0, with the law x²=x — this is Boolean algebra, exact and foundational, and (via Shannon) the literal basis of every digital gate.
His version was quirky. Boole used uninterpretable steps (x+x = 2x, even division). The clean modern algebra — where x+x = x — was tidied by Jevons, Peirce, Schröder after him.
He laid the stone. It's propositional/class logic only (no relations → Frege); and the circuit link that made it the world's logic is Shannon's (1937), seventy years on.
Set x and y to 1 (true) or 0 (false) and watch each logical operation compute as ordinary arithmetic — and land on exactly the right truth value. This is the whole revelation: "x and y" is literally x times y; "not x" is literally 1 minus x. Reasoning carried out as a sum.
What separates Boole's algebra from ordinary algebra is a single strange equation: x² = x In normal numbers that holds for almost nothing. In logic it must hold for everything — and the only numbers that satisfy it are 0 and 1. That one law is the whole confinement: it forces every quantity to be true or false, nothing in between. Its twin, x(1−x) = 0, is Aristotle's old law of non-contradiction, now written as arithmetic.
x(1−x)=0. Twenty-two centuries of logic, reduced to a quadratic."…the laws whose expression it is… are the laws of the human mind." — George Boole, The Laws of Thought (1854)
The leap that made Boole the most consequential mathematician you've half-heard-of came in 1937, when Claude Shannon realised an electrical switch is a 0 or a 1: open or closed. Wire two switches in series and current flows only if both close — that's AND, that's x·y. Wire them in parallel and it flows if either closes — that's OR. Flip the switches and watch the lamp obey Boole's arithmetic. Every chip in PSĒPHOS is this, a few billion times.
Gate kept on. Boole's own 1854 system was rougher than the clean thing taught today — he let expressions pass through meaningless intermediate values (2x, fractions) on the way to a sensible answer, and it was Jevons, Peirce and Schröder who filed it down to the idempotent lattice (x+x=x) we call Boolean algebra. His algebra is also propositional — it cannot say "every" or "some" or "taller than"; that needed Frege, next in this line. And Boole himself never touched a circuit: the marriage of his algebra to electricity is Shannon's, written when Boole had been dead seventy years. He died at forty-nine — pneumonia, after walking miles through rain to lecture, then being wrapped in wet sheets by a well-meaning remedy. Seen in his lifetime as an ingenious curiosity; revealed, a century later, as the man who wrote the grammar of the machine age.