This work develops a symbolic algebra of logic, representing propositions and classes with algebraic symbols and operations that mirror logical relations, formulates laws for combining and transforming logical expressions (analogues of conjunction, disjunction and negation), and shows how inference can be carried out algebraically; it also extends the approach to problems of probability and argues that thought follows lawful mathematical forms. The book established foundational ideas for what became Boolean algebra and had deep influence on formal logic, mathematics, and later developments in computing.