In Ancient India, the "Nasadiya Sukta" of the Rig Veda contains various logical divisions that were later recast formally as the four circles of catuskoti: "A", "not A", "A and not A" and "not A and not not A". The Nyaya school of Indian philosophical speculation is based on texts known as the "Nyaya Sutras" of Aksapada Gautama from around the 2nd Century B.C., and its methodology of inference is based on a system of logic (involving a combination of induction and deduction by moving from particular to particular via generality) that subsequently has been adopted by the majority of the other Indian schools.
But modern logic descends mainly from the Ancient Greek tradition. Both Plato and Aristotle conceived of logic as the study of argument and from a concern with the correctness of argumentation. Aristotle produced six works on logic, known collectively as the "Organon", the first of these, the "Prior Analytics", being the first explicit work in formal logic.
Aristotle espoused two principles of great importance in logic, the Law of Excluded Middle (that every statement is either true or false) and the Law of Non-Contradiction (confusingly, also known as the Law of Contradiction, that no statement is both true and false). He is perhaps most famous for introducing the syllogism (or term logic) (see the section on Deductive Logic below). His followers, known as the Peripatetics, further refined his work on logic.
In medieval times, Aristotelian logic (or dialectics) was studied, along with grammar and rhetoric, as one of the three main strands of the trivium, the foundation of a medieval liberal arts education.
Logic in Islamic philosophy also contributed to the development of modern logic, especially the development of Avicennian logic (which was responsible for the introduction of the hypothetical syllogism, temporal logic, modal logic and inductive logic) as an alternative to Aristotelian logic.
In the 18th Century, Immanuel Kant argued that logic should be conceived as the science of judgment, so that the valid inferences of logic follow from the structural features of judgments, although he still maintained that Aristotle had essentially said everything there was to say about logic as a discipline.
In the 20th Century, however, the work of Gottlob Frege, Alfred North Whitehead and Bertrand Russell on Symbolic Logic, turned Kant's assertion on its head. This new logic, expounded in their joint work "Principia Mathematica", is much broader in scope than Aristotelian logic, and even contains classical logic within it, albeit as a minor part. It resembles a mathematical calculus and deals with the relations of symbols to each other.