![]() Other philosophers who practised such dialectic reasoning were the so-called minor Socratics, including Euclid of Megara, who were probably followers of Parmenides and Zeno. Plato's Parmenides portrays Zeno as claiming to have written a book defending the monism of Parmenides by demonstrating the absurd consequence of assuming that there is plurality. This is the technique of drawing an obviously false, absurd or impossible conclusion from an assumption, thus demonstrating that the assumption is false. ![]() Separately from geometry, the idea of a standard argument pattern is found in the Reductio ad absurdum used by Zeno of Elea, a pre-Socratic philosopher of the fifth century BC. Fragments of early proofs are preserved in the works of Plato and Aristotle, and the idea of a deductive system was probably known in the Pythagorean school and the Platonic Academy. The three basic principles of geometry are that certain propositions must be accepted as true without demonstration, that all other propositions of the system are derived from these, and that the derivation must be formal, that is, independent of the particular subject matter in question. The systematic study of this seems to have begun with the school of Pythagoras in the late sixth century BC. While the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrative science. Esagil-kin-apli's medical Diagnostic Handbook in the 11th century BC was based on a logical set of axioms and assumptions, while Babylonian astronomers in the 8th and 7th centuries BC employed an internal logic within their predictive planetary systems, an important contribution to the philosophy of science. ![]() In particular, the ancient Egyptians had empirically discovered some truths of geometry, such as the formula for the volume of a truncated pyramid. It is probable that the idea of demonstrating a conclusion first arose in connection with geometry, which originally meant the same as "land measurement". However, logic studies the principles of valid reasoning, inference and demonstration. Valid reasoning has been employed in all periods of human history. The pyramids of Egypt were built using geometry Progress in mathematical logic in the first few decades of the twentieth century, particularly arising from the work of Gödel and Tarski, had a significant impact on analytic philosophy and philosophical logic, particularly from the 1950s onwards, in subjects such as modal logic, temporal logic, deontic logic, and relevance logic. The development of the modern so-called "symbolic" or "mathematical" logic during this period is the most significant in the two-thousand-year history of logic, and is arguably one of the most important and remarkable events in human intellectual history. Logic was revived in the mid-nineteenth century, at the beginning of a revolutionary period when the subject developed into a rigorous and formalistic discipline whose exemplar was the exact method of proof used in mathematics. The period between the fourteenth century and the beginning of the nineteenth century was largely one of decline and neglect, and is regarded as barren by at least one historian of logic. ![]() Greek logic, particularly Aristotelian logic, found wide application and acceptance in science and mathematics.Īristotle's logic was further developed by Islamic and Christian philosophers in the Middle Ages, reaching a high point in the mid-fourteenth century. Formal logic was developed in ancient times in China, India, and Greece. ![]() The history of logic is the study of the development of the science of valid inference ( logic). ![]()
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