Book: The Mathematical Analysis of Logic

Overview
George Boole's 1847 treatise, The Mathematical Analysis of Logic, recasts logic as a branch of mathematics by introducing a symbolic calculus for reasoning. Boole argues that logical relations among classes and propositions can be expressed and manipulated using algebraic symbols, and that valid inference follows from formal algebraic operations. The work marks a decisive move away from purely syllogistic treatments toward a system that treats logical forms as mathematical objects.

Core Concepts
Boole distinguishes between the mental operations of selection and exclusion and their representation by symbols he calls "elective symbols." These symbols denote operations that pick out members of classes; their algebraic combinations correspond to logical combinations. Central to the system are the notions of universal and empty classes, represented by the constants 1 and 0, and the idea that an elective symbol is idempotent, so that x^2 = x, reflecting that selecting twice is the same as selecting once.

Algebraic Method
The heart of Boole's method is treating propositions as equations and using algebraic manipulation to derive consequences. He adopts familiar algebraic laws such as commutativity, associativity, and distributivity, but modifies their interpretation to fit logical subjects. Solving logical problems becomes analogous to solving algebraic equations: unknown classes or relations are isolated by reduction, and solutions yield the class expressions that satisfy given premises. Boole also introduces elimination techniques that permit the removal of intermediate terms to obtain direct relations among remaining symbols.

Notation and Logical Operations
Operations that later became standard in Boolean algebra are foreshadowed here. Multiplication represents conjunction or intersection of classes, addition represents a disjunctive combination subject to caution about overlaps, and complement is treated as an operation related to subtraction from the universal class. Boole's algebra differs from ordinary algebra chiefly because of the idempotent law and the constrained numerical values assigned to symbols, which are not arbitrary numbers but values reflecting membership relations. He develops methods to translate ordinary propositions into algebraic equations and to interpret algebraic solutions back into logical statements.

Philosophical Foundations
Boole grounds his technical calculus in a philosophical claim that logic should be deductive and exact like mathematics. He views the laws of thought as amenable to quantitative treatment and insists that symbolic expression reveals the structure underlying ordinary logical discourse. Rather than providing a mere mnemonic for syllogisms, Boole seeks a general theory that can handle complex quantified statements, multiple classes, and relations not easily captured by traditional logic.

Legacy and Influence
Though concise, the 1847 book laid the foundation for the algebraic study of logic that culminated in Boole's later An Investigation of the Laws of Thought (1854). Subsequent mathematicians and logicians refined his notation and resolved ambiguities in interpretation, leading to the modern field of Boolean algebra. The algebraic paradigm influenced work by De Morgan, Schröder, Peirce, and ultimately the symbolic logicians of the late 19th and 20th centuries, and it became central to digital circuit design and theoretical computer science. Boole's insistence that logical reasoning could be formalized and computed remains one of the most decisive shifts in the intellectual history of logic.