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Logic

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(Allegory of logic)

The logic, from the Greek λογική / logikê, is a term derived from λόγος / lógos – meaning at the same time “reason”, “language”, and “reasoning” – is, in a first approach, the study of the formal rules that must respect all correct arguments. The term would have been used for the first time by Xenocrates.

It has been since ancient times one of the great disciplines of philosophy, with ethics (moral philosophy) and physics (science of nature). In the Middle Ages, it does not appear explicitly among the seven liberal arts (trivium: grammar, dialectic and rhetoric, quadrivium: arithmetic, geometry, astronomy and music). In addition, since the 19th century (George Boole, Jevons), there has been a rapid development of a mathematical approach to logic. Its convergence with computer science since the end of the 20th century has given it renewed vitality. Since the 20th century, it has found numerous applications in engineering, linguistics, cognitive psychology, analytic philosophy and communication. Ancient logic breaks down into dialectic, rhetoric.

History

Antiquity

Logic is at the origin the search for general and formal rules making it possible to distinguish a conclusive reasoning from that which is not it. She found her first gropings in mathematics and especially in geometry but it was mainly under the leadership of the Megarics and then Aristotle that she took off.

Logic was used very early against itself, that is to say against the very conditions of speech: the sophist Gorgias uses it in his Treatise on non-being in order to prove that there is no possible ontology: “it is not being that is the object of our thoughts”: the material truth of logic is thus ruined. Language thus acquires its own law, that of logic, independent of reality. But the sophists have been removed from the history of philosophy (sophist has taken a pejorative meaning), so that logic, in the understanding that we had for example in the Middle Ages, remained subject to the thought of being.

Contemporary era

In the seventeenth century, Leibniz made fundamental research in logic that profoundly revolutionized the logic of Aristotle. He constantly claims the tradition of Aristotle’s syllogisms and tries to integrate it into his own system. He is the first to imagine and develop a completely formal logic.

Emmanuel Kant, on the other hand, defines logic as “a science that exposes in detail and rigorously demonstrates the formal rules of all thought.” The six works of Aristotle grouped under the title of Organon, which notably include the Categories and the study of the syllogism, were long considered the reference on this subject. In 1847, George Boole’s book, Mathematical Analysis of Logic was published, followed by An Investigation Into the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities. Boole develops a new form of logic, both symbolic and mathematical. Its purpose is to translate ideas and concepts into expressions and equations, to apply certain calculations to them and to translate the result into logical terms, thus marking the beginning of modern logic, based on an algebraic and semantic approach, which we later called Boolean algebra in his honor.

The different approaches

In very general terms there are four approaches to logic:

  • History: we are interested in the evolution and development of logic and especially in the Aristotelian syllogistics and attempts since Leibniz to make logic a true algorithmic calculation. This historical approach is particularly interesting for philosophy because both Aristotle and the Stoics or Leibniz have worked as philosophers and logicians.
  • Mathematics: Contemporary mathematical logic is related to mathematics, computer science and engineering.
  • Philosophy: Philosophy and above all analytic philosophy, which essentially studies propositional language, are based on analytical and argumentative tools derived, on the one hand, from the logical developments made during the history of philosophy, and on the other hand from recent developments in mathematical logic. On the other hand, philosophy and especially the philosophy of logic set themselves the task of enlightening the fundamental concepts and methods of logic.
  • Computing: Attacking the automation of calculations and demonstrations, the theoretical foundations of systems design, programming and artificial intelligence. The computer approach is crucial today because it is by trying to mechanize the reasoning, or even to automate it, that logic and mathematics are experiencing a real revolution since the end of the 20th century following the exploitation of proof-program correspondence. The epistemological consequences of these developments are still largely unsuspected.

The mathematical approach has a position that is somewhat peculiar from an epistemological point of view, since it is both a tool for defining mathematics, and a branch of these same mathematics, and therefore an object.

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