section_path: History
section_name: History
paragraph_text: History
  
  Logic was developed independently in several cultures during antiquity. One major early contributor was Aristotle, who developed term logic in his Organon and Prior Analytics. He was responsible for the introduction of the hypothetical syllogism and temporal modal logic. Further innovations include inductive logic as well as the discussion of new logical concepts such as terms, predicables, syllogisms, and propositions. Aristotelian logic was highly regarded in classical and medieval times, both in Europe and the Middle East. It remained in wide use in the West until the early 19th century. It has now been superseded by later work, though many of its key insights are still present in modern systems of logic.
  
  Ibn Sina (Avicenna) was the founder of Avicennian logic, which replaced Aristotelian logic as the dominant system of logic in the Islamic world. It influenced Western medieval writers such as Albertus Magnus and William of Ockham. Ibn Sina wrote on the hypothetical syllogism and on the propositional calculus. He developed an original "temporally modalized" syllogistic theory, involving temporal logic and modal logic. He also made use of inductive logic, such as his methods of agreement, difference, and concomitant variation, which are critical to the scientific method. Fakhr al-Din al-Razi was another influential Muslim logician. He criticized Aristotelian syllogistics and formulated an early system of inductive logic, foreshadowing the system of inductive logic developed by John Stuart Mill.
  
  During the Middle Ages, many translations and interpretations of Aristotelian logic were made. The works of Boethius were particularly influential. Besides translating Aristotle's work into Latin, he also produced textbooks on logic. Later, the works of Islamic philosophers such as Ibn Sina and Ibn Rushd (Averroes) were drawn on. This expanded the range of ancient works available to medieval Christian scholars since more Greek work was available to Muslim scholars that had been preserved in Latin commentaries. In 1323, William of Ockham's influential Summa Logicae was released. It is a comprehensive treatise on logic that discusses many basic concepts of logic and provides a systematic exposition of types of propositions and their truth conditions.
  
  In Chinese philosophy, the School of Names and Mohism were particularly influential. The School of Names focused on the use of language and on paradoxes. For example, Gongsun Long proposed the white horse paradox, which defends the thesis that a white horse is not a horse. The school of Mohism also acknowledged the importance of language for logic and tried to relate the ideas in these fields to the realm of ethics.
  
  In India, the study of logic was primarily pursued by the schools of Nyaya, Buddhism, and Jainism. It was not treated as a separate academic discipline and discussions of its topics usually happened in the context of epistemology and theories of dialogue or argumentation. In Nyaya, inference is understood as a source of knowledge (pramāṇa). It follows the perception of an object and tries to arrive at conclusions, for example, about the cause of this object. A similar emphasis on the relation to epistemology is also found in Buddhist and Jainist schools of logic, where inference is used to expand the knowledge gained through other sources. Some of the later theories of Nyaya, belonging to the Navya-Nyāya school, resemble modern forms of logic, such as Gottlob Frege's distinction between sense and reference and his definition of number.
  
  The syllogistic logic developed by Aristotle predominated in the West until the mid-19th century, when interest in the foundations of mathematics stimulated the development of modern symbolic logic. Many see Gottlob Frege's Begriffsschrift as the birthplace of modern logic. Gottfried Wilhelm Leibniz 's idea of a universal formal language is often considered a forerunner. Other pioneers were George Boole, who invented Boolean algebra as a mathematical system of logic, and Charles Peirce, who developed the logic of relatives. Alfred North Whitehead and Bertrand Russell, in turn, condensed many of these insights in their work Principia Mathematica. Modern logic introduced novel concepts, such as functions, quantifiers, and relational predicates. A hallmark of modern symbolic logic is its use of formal language to precisely codify its insights. In this regard, it departs from earlier logicians, who relied mainly on natural language. Of particular influence was the development of first-order logic, which is usually treated as the standard system of modern logic. Its analytical generality allowed the formalization of mathematics and drove the investigation of set theory. It also made Alfred Tarski's approach to model theory possible and provided the foundation of modern mathematical logic.