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Logic in India Main article: Indian logic Logic began independently in ancient India and continued to develop through to early modern times, without any known influence from Greek logic.[30] Medhatithi Gautama (c. 6th century BCE) founded the anviksiki school of logic.[31] The Mahabharata (12.173.45), around the 5th century BCE, refers to the anviksiki and tarka schools of logic. Pāṇini (c. 5th century BCE) developed a form of logic (to which Boolean logic has some similarities) for his formulation of Sanskrit grammar. Logic is described by Chanakya (c. 350-283 BCE) in his Arthashastra as an independent field of inquiry anviksiki.[32] Two of the six Indian schools of thought deal with logic: Nyaya and Vaisheshika. The Nyaya Sutras of Aksapada Gautama (c. 2nd century CE) constitute the core texts of the Nyaya school, one of the six orthodox schools of Hindu philosophy. This realist school developed a rigid five-member schema of inference involving an initial premise, a reason, an example, an application and a conclusion.[33] The idealist Buddhist philosophy became the chief opponent to the Naiyayikas. Nagarjuna (c. 150-250 CE), the founder of the Madhyamika ("Middle Way") developed an analysis known as the catuskoti (Sanskrit). 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 (c 480-540 CE), who is sometimes said to have developed a formal syllogistic,[34] and his successor Dharmakirti that Buddhist logic reached its height. (Whether their analysis actually constitutes a formal syllogistic is contested.) Their analysis centered on the definition of an inference-warranting relation, "vyapti", also known as invariable concomitance or pervasion.[35] To this end a doctrine known as "apoha" or differentiation was developed.[36] 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-Nyāya, which developed a formal analysis of inference in the sixteenth century. This later school began around eastern India and Bengal, and developed theories resembling modern logic, such as Gottlob Frege's "distinction between sense and reference of proper names" and his "definition of number," as well as the Navya-Nyaya theory of "restrictive conditions for universals" anticipating some of the developments in modern set theory.[37] Since 1824, Indian logic attracted the attention of many Western scholars, and has had an influence on important 19th-century logicians such as Charles Babbage, Augustus De Morgan, and particularly George Boole, as confirmed by his wife Mary Everest Boole who wrote in an "open letter to Dr Bose" titled "Indian Thought and Western Science in the Nineteenth Century" written in 1901:[38][39] "Think what must have been the effect of the intense Hinduizing of three such men as Babbage, De Morgan and George Boole on the mathematical atmosphere of 1830-1865" As a general matter, some authorities have maintained that classical Indian logic is not actually a theory of deductive validity and that the tendency to suppose otherwise is the effect of imposing Western expectations on the ancient material. Bimal Krishna Matilal remarks, “The usual distinction, so well entrenched in the Western tradition, between deduction and induction was not to be found in the same way in the Indian tradition. The argument patterns studied were at best an unconscious mix of the two processes.” He continues, “The inductive element of the argument patterns studied by the Indian philosophers has thus often been lost sight of by modern scholars who emphasize the alleged certainty of the inferred conclusions, and then go on to equate the Indian argument patterns invariably with deductive or syllogistic forms.”[40] On this view, though Indian logic aimed to establish the grounds of reasonable inference, it did not distinguish between the logically necessary inferences of valid deduction and the highly probable inferences of reasonable induction. Matilal argues that ancient Indian philosophers were largely uninterested in formal deductive validity for its own sake, and in this sense “Indian logic is not formal logic.”[41] Seen in this way, Dignāga’s famous “wheel of reason” (Hetucakra) is a method of indicating when one thing (such as smoke) can be taken as an invariable sign of another thing (like fire), but the inference is often inductive and based on past observation. Matilal remarks that Dignāga’s analysis is much like John Stuart Mill’s Joint Method of Agreement and Difference, which is inductive.[42] In addition, the traditional five-member Indian syllogism, though deductively valid, has repetitions that are unnecessary to its logical validity. As a result, some commentators see the traditional Indian syllogism as a rhetorical form that is entirely natural in many cultures of the world, and yet not as a logical form—not in the sense that all logically unnecessary elements have been omitted, for the sake of analysis. 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. 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. |
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