An Investigation of the Laws of Thought
George Boole · 1854 · Project Gutenberg
Full title: An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities.
Reduced logic to algebra. Propositions become equations, reasoning becomes calculation. Every circuit, every if-statement, every formal proof descends from this book.
Logic as algebra
Boole's insight was that logical propositions obey the same laws as algebraic equations. "All X are Y" becomes x(1-y) = 0. "Some X are Y" becomes xy ≠ 0. Syllogisms that Aristotle resolved by pattern-matching, Boole resolved by solving equations. The move sounds simple. It changed everything.
What it enabled
Boole published in 1854. Eighty years later, 
Claude Shannon showed in his 1937 master's thesis that Boolean algebra maps directly to electrical switching circuits. AND becomes a series circuit. OR becomes a parallel circuit. NOT becomes an inverter. Every digital computer is a physical instantiation of Boole's algebra. The gap between "logic is algebra" and "logic is hardware" turned out to be one engineering thesis.
Programming inherited the same structure. Every conditional branch, every logical operator, every type-checker evaluates Boolean expressions. When you write if (x && !y), you are writing Boole's notation with different symbols. The abstraction has survived 170 years without modification because it was right the first time.
Formal verification rests on the same foundation. To prove a program correct, you express its specification as logical propositions and its behavior as logical propositions, then check that the behavior implies the specification. The checking is algebraic manipulation of Boolean expressions. Boole did not anticipate computers, but he built the algebra they would need.
Connections
Peirce extended Boole's propositional logic into predicate logic and quantification, adding "for all" and "there exists" to Boolean "and," "or," and "not." Without that extension, Boole's system could handle classes but not relations. Peirce also drew the link between logic and pragmatism: if reasoning is computation, then the meaning of a proposition is the set of operations it licenses.
Popper's falsifiability depends on formal logic in a specific way. A scientific theory is falsifiable when its negation is satisfiable: there exists some observation that would make the theory false. Expressing this precisely requires the machinery Boole built. Popper could not have drawn his demarcation line without a formal account of negation and logical consequence.
The logic section of this reading site traces the line forward from Boole through Frege, Russell, Godel, and Turing. Boole is the first step. Every subsequent development either extends his algebra or discovers its limits.
The integrity angle
Boole's contribution to intellectual integrity is underappreciated. Before Boolean algebra, logical arguments were evaluated by intuition, rhetorical skill, and pattern-matching against Aristotle's syllogistic forms. A clever sophist could construct arguments that felt valid but were not. Boole made validity checkable. Reduce the argument to algebra, solve the equations, and the answer is mechanical. No room for persuasion to override proof.
This matters for science because scientific reasoning is a chain of inferences. If any link in the chain is invalid, the conclusion does not follow. Boole gave scientists a way to audit their own reasoning by translating it into a form where errors become visible. Mill gave them methods for extracting causes from observations. Boole gave them methods for checking whether their conclusions actually follow from their premises.