We shall review the development of logic, beginning with Aristotle's syllogistic logic. Next the tradition of algebraic logic is described, through the work of George Boole, Ernst Schröder, and Thoralf Skolem. There follows a section on axiomatic logic with two phases, the early work of Gottlob Frege, Giuseppe Peano, and Bertrand Russell, and a second phase with David Hilbert and Paul Bernays, and up to Heyting's intuitionistic logic in 1930. That is the point right before Gentzen's development of natural deduction and sequent calculus. The emphasis is on how the logical systems work, i.e., on their deductive machinery.

Aristotle's deductive logic

Aristotle's system of deductive logic, also known as the ‘theory of syllogisms’, has been interpreted in various ways in the long time since it was conceived. The situation is not different from the reading of other chapters of the formal sciences of antiquity, such as Euclid's geometry and works of Archimedes. When Frege invented predicate logic, he finished the presentation proudly with a reconstruction of the Aristotelian forms of propositions, such as Every A is B that is interpreted as ∀x (A(x) ⊃ B (x)), with a universal quantification over some domain and the predicates A and B. Frege reproduced similarly Aristotelian inferences, such as the conclusion Every A is C obtained from the premisses Every A is B and Every B is C, in the way shown in Section 9.1.