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The branch of philosophy and mathematics concerned with analyzing the rules that govern correct and incorrect reasoning, or inference. It was created by Aristotle, who analyzed terms and propositions and in his Prior Analytics set out systematically the various forms of syllogism; this work has remained an important part of logic ever since. Aristotle's other great achievement was the use of symbols to expose the form of an argument independently of its content. Thus a typical Aristotelian syllogism might be: all A is B; all B is C; therefore all A is C. This formalization of arguments is fundamental to all logic.

Aristotle's pupil Theophrastus developed syllogistic logic, and soma of the Stoics used symbols to represent not single terms but whole propositions, but apart from this there were no significant developments in later antiquity or the early Middle Ages, although logic (dialectic) was part of the trivium. From the 12th century onward there was greater revival of interest in logic: Latin translations of Aristotle's logical works (collectively called the Organun) were studied intently, and a kind of program emerged, which was based on Aristotle and included much that would nowadays be regarded as grammar, epistemology, and linguistic analysis. This Scholastic period was a great age of commentaries and compendiums, with much refinement and minute analysis but little original work. Among the most important medieval logicians were William of Ockham, Albert of Saxony, and Jean Buridan.

After the Renaissance an anti-Aristotelian reaction set in, and logic was given a new turn by Petrus Ramus and by Francis Bacon's prescription that induction (and not deduction) should be the method of the new science. In the work of George Boole and Gottlob Frege the 19th century saw a vast extension in the scope and power of logic. In particular logic became as bound up with mathematics as it was with philosophy. Logicians became interested in whether particular logical systems were either consistent or complete. (A consistent logic is one in which contradictory statements cannot be validly derived.) The climax of 20th-century logic came in the early 1930s when Kurt Gödel demonstrated both the completeness of Frege's first-order logic and that no higher-order logic could be both consistent and complete.

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