History of logic
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The history of logic documents the development of logic as it occurs in various rival cultures and traditions in history. While many cultures have employed intricate systems of reasoning, logic as an explicit analysis of the methods of reasoning received sustained development originally only in 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 BC. The formally sophisticated treatment of modern logic descends from the Greek tradition, but comes not wholly through Europe, but instead comes from the transmission of Aristotelian logic and commentary upon it by Islamic philosophers to Medieval European logicians.
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Logic in China
- Main article: Logic in China
In China, a contemporary of Confucius, Mozi, "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.
Logic in India
- Main article: Indian logic
The Nyaya Sutras of Aksapada Gautama constitute the core texts of the Nyaya school, one of the six orthodox schools of Hindu 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, which introduced a formal analysis of inference in the 16th century.
Logic in Greece
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.
Logic in Islamic philosophy
- Main article: Logic in Islamic philosophy
For after Muhammed's death, Islamic law placed importance on formulating standards of argument, which gave rise to a novel approach to argumentation in Kalam, but this approach was displaced by ideas from Greek philosophy with the rise of the Mutazilite philosophers, who valued highly Aristotle's Organon. The work of Greek influenced Islamic philosophers were crucial in the reception of Greek logic in medieval Europe, and the commentaries on the Organon by Averroes played a central role in the subsequent flowering of medieval European logic.
Despite the logical sophistication of Al-Ghazali, the rise of the Asharite school slowly suffocated original work on logic in the Islamic world.
Medieval logic
Descartes and the Port-Royal logic
The Port Royal logic ("L'Art de penser", published 1662) defines logic as "the art of using reason well in the acquisition of the knowledge of things, both for one's own instruction and that of others." More briefly "Logic is the art of reasoning." The latter is Arnauld's definition. Definitions of this type are considered too narrow, both because they define logic in terms of art, not leaving room for its claim to be considered a science, and because, by the use of the term reasoning, they restrict the scope of logic to one class of mental processes.
The advent of modern 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
References
- Dov Gabbay and John Woods, eds. Handbook of the history of logic: Volume 1: Greek, Indian and Arabic logic. Elsevier, Amsterdam, 2004.
External links
- Annotated bibliography on the history of logic (http://www.formalontology.it/history_of_logic.htm)
- Overview of Indian and Greek Development of Logic and Language (http://en.wikipedia.org/wiki/User:Hbmartin/Overview_of_Indian_and_Greek_Development_of_Logic_and_Language)