How did George Boole use combinatorial methods in symbolic logic?

George Boole utilized combinatorial methods to advance symbolic logic by treating logical propositions as algebraic variables, a system now known as Boolean algebra.123

Rather than focusing on the specific content of propositions, Boole applied the laws of combination to these symbols, allowing for the systematic derivation of conclusions from premises. His approach involved:42

• Algebraic Representation: He used symbols to represent classes of "things," where the universal class was denoted by "1" and the null class by "0".4
• Logical Operations: He employed algebraic operations to represent logical connectives. For instance, the juxtaposition of terms (e.g., "AB") represented the intersection of classes, while addition signified the non-overlapping union of classes.4
• Expansion and Inference: Boole extensively explored equations and techniques, including binomial-like expansions, to manipulate these logical expressions. This allowed him to treat deductive logic as a calculus, where the validity of the process depended solely on the laws of combination rather than the interpretation of the symbols themselves.42

By bridging the gap between algebraic analysis and logic, Boole provided a formal framework that remains fundamental to information theory and the design of digital computers.12