While many cultures employ intricate systems of reasoning, it is generally agreed that logic as an explicit analysis of the methods of reasoning was independently developed by only three traditions: China, India and Greece. Although exact dates are uncertain, especially in the case of India, it is possible that logic emerged in all three societies in the 4th century BCE.

In China, a contemporary of Confucius, Mo Zi, "Master Mo", is credited with founding the Mohist school, whose canons dealt with issues relating to valid inference and the conditions of correct conclusions. In particular, one of the schools that grew out of Mohism, the Logicians, are credited by some scholars for their early investigation of formal logic. Unfortunately, due to the harsh rule of Legalism in the subsequent Qin Dynasty, this line of investigation disappeared in China until the introduction of Indian philosophy by Buddhists.

The "Nyayasutras" of Gautama represent the basic texts of one of the six orthodox schools of Indian philosophy. This realist, one might say materialist, school worked out a rigid five-member schema of inference involving an initial premise, a reason, an example, an application and a conclusion. The idealist Buddhist philosophy became the chief opponent to the Naiyayikas. Nagarjuna, the founder of the Madhyamika "Middle Way" developed an analysis known as the "catuskoti" or tetralemma. This four-cornered argumentation systematically examined and rejected the affirmation of a proposition, its denial, the joint affirmation and denial, and finally, the rejection of its affirmation and denial. But it was with Dignaga and his successor Dharmakirti that Buddhist logic reached its height. Their analysis centered on the definition of necessary logical entailment, "vyapti", also known as invariable concomitance or pervasion. To this end a doctrine known as "apoha" or differentiation was developed. This involved what might be called inclusion and exclusion of defining properties. The difficulties involved in this enterprise, in part, stimulated the neo-scholastic school of Navya-Nyaya.

In Greece, Aristotle's collection of works known as the "Organon" or instrument almost ex nihilo created the discipline known as logic. Aristotle's examination of the syllogism bears interesting comparison with the Indian schema of inference and the less rigid Chinese discussion. Through Latin in Western Europe, and disparate languages more to the East, such as Arabic, Armenian and Georgian, the Aristotelian tradition was considered to pre-eminently codify the laws of reasoning. It was only in the Nineteenth Century that acquaintance with the classical literature of India and deeper knowledge of China brought about a change in this viewpoint.

Formal logic

Historically, Descartes, may have been the first philosopher to have had the idea of using algebra, especially its techniques for solving for unknown quantities in equations, as a vehicle for scientific exploration. However, the idea of a calculus of reasoning was cultivated especially by Gottfried Wilhelm Leibniz. Though modern logic in its present form originates with Boole and De Morgan, Leibniz was the first to have a really distinct plan of a broadly applicable system of mathematical logic. That this is so appears from research - much of which is quite recent - into Leibniz's unpublished work.

Gottlob Frege in his 1879 Begriffsschrift extended formal logic beyond propositional logic to include constructors such as "all", "some". He showed how to introduce variables and quantifiers to reveal the logical structure of sentences, which may have been obscured by their grammatical structure. For instance, "All humans are mortal" becomes "All things x are such that, if x is a human then x is mortal."

Charles Peirce introduced the term "second-order logic" and provided us with most of our modern logical notation, including the symbols ∀ and ∃. Although Peirce published his work some time after the Begriffsschrift, Frege's contribution wasn't very well known until many years later. Logicians in the late 19th and early 20th centuries were thus more familiar with Peirce's system of logic (although Frege is generally recognized today as being the "Father of modern logic").

In 1889 Giuseppe Peano published the first version of the logical axiomatization of arithmetic. Five of the nine axioms he came up with are now known as the Peano axioms. One of these axioms was a formalized statement of the principle of mathematical induction.

See also

Article Term Logic