Boolean logic

Boolean logic is arithmetic for truth. Every variable holds one of two values — true or false, 1 or 0 — and three operations combine them: AND (true only if both inputs are true), OR (true if either is true), and NOT (flips true to false). From those three, any logical statement whatsoever can be built and, crucially, simplified by algebraic rules: a tangled condition can be reduced to its smallest equivalent form the way `2x + 3x` reduces to `5x`. Boole laid it out in an 1847 pamphlet and in full in his 1854 book *An Investigation of the Laws of Thought*, proposing that the laws of human reasoning could be written as equations.

For two thousand years logic had been Aristotle's — a catalogue of valid argument forms studied in words. Boole made it mathematics: something you could compute with, prove things about, and mechanize. That last part is everything. A machine cannot follow a rhetorical syllogism, but it can very easily be true or false, on or off, and combine those states by fixed rules. Boolean logic is the bridge that let physical devices — relays, then vacuum tubes, then transistors — carry out reasoning, because it reduced reasoning to operations a switch can perform.

Every digital device is a physical Boolean expression. Logic gates, adders, memory cells, processors, and the entire discipline of digital design are built from AND/OR/NOT acting on 1s and 0s. Boolean algebra also underlies database queries, search engines (AND/OR/NOT keywords), and the conditionals in every program. Boole died in 1864 believing he had formalized the laws of thought; he had, in fact, written the operating manual for machines that would not exist for another eighty years.