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59-606: The tetralemma is a figure that features prominently in the logic of India . It states that with reference to any a logical proposition (or axiom) X, there are four possibilities: The history of fourfold negation, the Catuskoti (Sanskrit), is evident in the logico-epistemological tradition of India, given the categorical nomenclature Indian logic in Western discourse. Subsumed within the auspice of Indian logic, ' Buddhist logic ' has been particularly focused in its employment of

118-453: A grammatical predicate, as in the sentence "the person coming this way is Callias". But it is still a logical subject. He contrasts universal ( katholou ) secondary substance, genera, with primary substance, particular ( kath' hekaston ) specimens. The formal nature of universals , in so far as they can be generalized "always, or for the most part", is the subject matter of both scientific study and formal logic. The essential feature of

177-488: A part of Catholic theological reasoning. For example, Joyce's Principles of Logic (1908; 3rd edition 1949), written for use in Catholic seminaries, made no mention of Frege or of Bertrand Russell . Some philosophers have complained that predicate logic: Even academic philosophers entirely in the mainstream, such as Gareth Evans , have written as follows: George Boole 's unwavering acceptance of Aristotle's logic

236-453: A result, Nyaya scholars again went to great pains to identify, in each case, what it took to make knowledge valid, in the process creating a number of explanatory schemes. In this sense, Nyaya is probably the closest Indian equivalent to contemporary analytic philosophy . Jainism made its own unique contribution to this mainstream development of logic by also occupying itself with the basic epistemological issues, namely, with those concerning

295-544: A set of thorough criticisms of Nyāya theories of thought and language. In his book, Gangeśa both addressed some of those criticisms and – more importantly – critically examined the Nyāya darśana himself. He held that, while Śrīharśa had failed successfully to challenge the Nyāya realist ontology, his and Gangeśa's own criticisms brought out a need to improve and refine the logical and linguistic tools of Nyāya thought, to make them more rigorous and precise. Tattvacintāmani dealt with all

354-440: A word. To assert "all Greeks are men" is not to say that the concept of Greeks is the concept of men, or that word "Greeks" is the word "men". A proposition cannot be built from real things or ideas, but it is not just meaningless words either. In term logic, a "proposition" is simply a form of language : a particular kind of sentence , in which the subject and predicate are combined, so as to assert something true or false. It

413-584: Is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, the Peripatetics . It was revived after the third century CE by Porphyry 's Isagoge . Term logic revived in medieval times, first in Islamic logic by Alpharabius in the tenth century, and later in Christian Europe in the twelfth century with

472-570: Is a member; for if we had any such universal, then, by hypothesis, we have got a given totality of all universals that exist and all of them belong to this big universal. But this universal is itself a universal and hence (since it cannot be a member of itself, because in Udayana's view no universal can be a member of itself) this universal too along with other universals must belong to a bigger universal and so on ad infinitum. What Udayana says here has interesting analogues in modern set theory in which it

531-678: Is a thought of the kind expressible by a declarative sentence) of a syllogism is a categorical sentence which has a subject and a predicate connected by a verb. The usual way of connecting the subject and predicate of a categorical sentence as Aristotle does in On Interpretation is by using a linking verb e.g. P is S. However, in the Prior Analytics Aristotle rejects the usual form in favour of three of his inventions: Aristotle does not explain why he introduces these innovative expressions but scholars conjecture that

590-595: Is an animal." Depending on the position of the middle term, Aristotle divides the syllogism into three kinds: syllogism in the first, second, and third figure. If the Middle Term is subject of one premise and predicate of the other, the premises are in the First Figure. If the Middle Term is predicate of both premises, the premises are in the Second Figure. If the Middle Term is subject of both premises,

649-514: Is assumed, quantification implies the existence of at least one subject, unless disclaimed. For Aristotle, the distinction between singular and universal is a fundamental metaphysical one, and not merely grammatical . A singular term for Aristotle is primary substance , which can only be predicated of itself: (this) "Callias" or (this) "Socrates" are not predicable of any other thing, thus one does not say every Socrates one says every human ( De Int. 7; Meta. D9, 1018a4). It may feature as

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708-474: Is clearly awkward, a weakness exploited by Frege in his devastating attack on the system. The famous syllogism "Socrates is a man ...", is frequently quoted as though from Aristotle, but in fact, it is nowhere in the Organon . Sextus Empiricus in his Hyp. Pyrrh (Outlines of Pyrronism) ii. 164 first mentions the related syllogism "Socrates is a human being, Every human being is an animal, Therefore, Socrates

767-512: Is emphasized by the historian of logic John Corcoran in an accessible introduction to Laws of Thought Corcoran also wrote a point-by-point comparison of Prior Analytics and Laws of Thought . According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were “to go under, over, and beyond” Aristotle's logic by: More specifically, Boole agreed with what Aristotle said; Boole's ‘disagreements’, if they might be called that, concern what Aristotle did not say. First, in

826-703: Is evidence that Aristotle knew of fourth-figure syllogisms. In the Prior Analytics translated by A. J. Jenkins as it appears in volume 8 of the Great Books of the Western World, Aristotle says of the First Figure: "... If A is predicated of all B, and B of all C, A must be predicated of all C." In the Prior Analytics translated by Robin Smith, Aristotle says of the first figure: "... For if A

885-646: Is held that a set of all sets (i.e., a set to which every set belongs) does not exist. In the late 18th-century British scholars began to take an interest in Indian philosophy and discovered the sophistication of the Indian study of inference. This process culminated in Henry T. Colebrooke's The Philosophy of the Hindus: On the Nyaya and Vaisesika Systems in 1824, which provided an analysis of inference and comparison to

944-432: Is mortal is affirmative, since the mortality of philosophers is affirmed universally, whereas no philosopher is mortal is negative by denying such mortality in particular. The quantity of a proposition is whether it is universal (the predicate is affirmed or denied of all subjects or of "the whole") or particular (the predicate is affirmed or denied of some subject or a "part" thereof). In case where existential import

1003-531: Is not a thought, or an abstract entity . The word "propositio" is from the Latin, meaning the first premise of a syllogism . Aristotle uses the word premise ( protasis ) as a sentence affirming or denying one thing or another ( Posterior Analytics 1. 1 24a 16), so a premise is also a form of words. However, as in modern philosophical logic, it means that which is asserted by the sentence. Writers before Frege and Russell , such as Bradley , sometimes spoke of

1062-465: Is predicated of every B and B of every C, it is necessary for A to be predicated of every C." Taking a = is predicated of all = is predicated of every , and using the symbolical method used in the Middle Ages, then the first figure is simplified to: Or what amounts to the same thing: When the four syllogistic propositions, a, e, i, o are placed in the first figure, Aristotle comes up with

1121-427: Is the basic component of the proposition. The original meaning of the horos (and also of the Latin terminus ) is "extreme" or "boundary". The two terms lie on the outside of the proposition, joined by the act of affirmation or denial. For early modern logicians like Arnauld (whose Port-Royal Logic was the best-known text of his day), it is a psychological entity like an "idea" or " concept ". Mill considers it

1180-411: Is the name for a largely comparable, but not equatable, 'four corner argument' within the tradition of Classical logic . Nyāya ( ni-āyá , literally "recursion", used in the sense of " syllogism , inference") is the name given to one of the six orthodox or astika schools of Hindu philosophy — specifically the school of logic. The Nyaya school of philosophical speculation is based on texts known as

1239-546: The Nyaya Sutras , which were written by Gotama in around the 2nd century CE. The most important contribution made by the Nyaya school to modern Hindu thought is its methodology. This methodology is based on a system of logic that has subsequently been adopted by most of the other Indian schools (orthodox or not), much in the same way that Western philosophy can be said to be largely based on Aristotelian logic . Followers of Nyaya believed that obtaining valid knowledge

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1298-504: The Organon . Two of these texts in particular, namely the Prior Analytics and De Interpretatione , contain the heart of Aristotle's treatment of judgements and formal inference , and it is principally this part of Aristotle's works that is about term logic . Modern work on Aristotle's logic builds on the tradition started in 1951 with the establishment by Jan Lukasiewicz of a revolutionary paradigm. Lukasiewicz's approach

1357-457: The Rigveda ( RV 10 .129) contains ontological speculation in terms of various logical divisions that were later recast formally as the four circles of catuskoti : "A", "not A", "A and 'not A'", and "not A and not not A". Medhatithi Gautama (c. 6th century BCE) founded the anviksiki school of logic. The Mahabharata (12.173.45), around the 4th century BCE to 4th century CE, refers to

1416-541: The Prior Analytics , Aristotle identifies valid and invalid forms of arguments called syllogisms. A syllogism is an argument that consists of at least three sentences: at least two premises and a conclusion. Although Aristotle does not call them " categorical sentences", tradition does; he deals with them briefly in the Analytics and more extensively in On Interpretation . Each proposition (statement that

1475-468: 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 . Vaisheshika, also Vaisesika, (Sanskrit: वैशेषिक) is one of the six Hindu schools of Indian philosophy . It came to be closely associated with

1534-427: The syllogism is that, of the four terms in the two premises, one must occur twice. Thus The subject of one premise, must be the predicate of the other, and so it is necessary to eliminate from the logic any terms which cannot function both as subject and predicate, namely singular terms. However, in a popular 17th-century version of the syllogism, Port-Royal Logic , singular terms were treated as universals: This

1593-422: The "judgment" as something distinct from a sentence, but this is not quite the same. As a further confusion the word "sentence" derives from the Latin, meaning an opinion or judgment , and so is equivalent to " proposition ". The logical quality of a proposition is whether it is affirmative (the predicate is affirmed of the subject) or negative (the predicate is denied of the subject). Thus every philosopher

1652-761: The Greek view and followed a course which seems to have originated in India and which has been transmitted, with additions, to us by the Arabs; in it the concept of number appears as logically prior to the concepts of geometry. [...] But the present trend in mathematics is clearly in the direction of a return to the Greek standpoint; we now look upon each branch of mathematics as determining its own characteristic domain of quantities." Aristotelian logic In logic and formal semantics , term logic , also known as traditional logic , syllogistic logic or Aristotelian logic ,

1711-548: The Hindu school of logic, Nyaya. Vaisheshika espouses a form of atomism and postulates that all objects in the physical universe are reducible to a finite number of atoms. Originally proposed by Kanāda (or Kana-bhuk, literally, atom-eater) from around the 2nd century BCE. In the 2nd century, the Buddhist philosopher Nagarjuna refined the Catuskoti form of logic. The Catuskoti is also often glossed Tetralemma (Greek) which

1770-624: The Middle Ages, for mnemonic reasons, these six forms were called respectively: "Darapti", "Felapton", "Disamis", "Datisi", "Bocardo" and "Ferison". Term logic began to decline in Europe during the Renaissance , when logicians like Rodolphus Agricola Phrisius (1444–1485) and Ramus (1515–1572) began to promote place logics. The logical tradition called Port-Royal Logic , or sometimes "traditional logic", saw propositions as combinations of ideas rather than of terms, but otherwise followed many of

1829-529: The Navya-Nyaya theory of "restrictive conditions for universals" anticipating some of the developments in modern set theory . Udayana in particular developed theories on "restrictive conditions for universals" and " infinite regress" that anticipated aspects of modern set theory. According to Kisor Kumar Chakrabarti: In the third part we have shown how the study of the so-called 'restrictive conditions for universals' in Navya-Nyaya logic anticipated some of

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1888-644: The Nineteenth Century" written in 1901. De Morgan himself wrote in 1860 of the significance of Indian logic: "The two races which have founded the mathematics, those of the Sanskrit and Greek languages, have been the two which have independently formed systems of logic." Mathematicians became aware of the influence of Indian mathematics on the European. For example, Hermann Weyl wrote: "Occidental mathematics has in past centuries broken away from

1947-413: The Nyāya concepts into four main categories: sense or perception (pratyakşa), inference (anumāna), comparison or similarity ( upamāna ), and testimony (sound or word; śabda). 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

2006-525: The Prior Analytics, "... If one term belongs to all and another to none of the same thing, or if they both belong to all or none of it, I call such figure the third." Referring to universal terms, "... then when both P and R belongs to every S, it results of necessity that P will belong to some R." Simplifying: When the four syllogistic propositions, a, e, i, o are placed in the third figure, Aristotle develops six more valid forms of deduction: In

2065-408: The advent of new logic , remaining dominant until the advent of predicate logic in the late nineteenth century. However, even if eclipsed by newer logical systems, term logic still plays a significant role in the study of logic. Rather than radically breaking with term logic, modern logics typically expand it. Aristotle 's logical work is collected in the six texts that are collectively known as

2124-502: The ancient Indian philosophy , especially in the areas of skepticism and relativity. Following is the list of Jain philosophers who contributed to Jain Logic: Indian Buddhist logic (called Pramana ) flourished from about 500 CE up to 1300 CE. The three main authors of Buddhist logic are Vasubandhu (400–800 CE), Dignāga (480–540 CE), and Dharmakīrti (600–660 CE). The most important theoretical achievements are

2183-521: The conventions of term logic. It remained influential, especially in England, until the 19th century. Leibniz created a distinctive logical calculus , but nearly all of his work on logic remained unpublished and unremarked until Louis Couturat went through the Leibniz Nachlass around 1900, publishing his pioneering studies in logic. 19th-century attempts to algebraize logic, such as

2242-501: The developments of modern set theory. [...] In this section the discussion will center around some of the 'restrictive conditions for universals ( jatibadhaka ) proposed by Udayana. [...] Another restrictive condition is anavastha or vicious infinite regress. According to this restrictive condition, no universal ( jati ) can be admitted to exist, the admission of which would lead to a vicious infinite regress. As an example Udayana says that there can be no universal of which every universal

2301-539: The doctrine of Trairūpya (Skrt. त्रैरूप्य) and the highly formal scheme of the Hetucakra (Skrt. हेतुचक्र) ("Wheel of Reasons") given by Dignāga . There is still a vibrant living tradition of Buddhist logic in the Tibetan Buddhist traditions, where logic is an important part of the education of monks. The Navya-Nyāya or Neo-Logical darśana (school) of Indian philosophy was founded in the 13th century CE by

2360-518: The first figure has again come about)." The above statement can be simplified by using the symbolical method used in the Middle Ages: When the four syllogistic propositions, a, e, i, o are placed in the second figure, Aristotle comes up with the following valid forms of deduction for the second figure: In the Middle Ages, for mnemonic reasons they were called respectively "Camestres", "Cesare", "Festino" and "Baroco". Aristotle says in

2419-553: The first figure is axiomatic while the second and third require proof. The proof of the second and third figure always leads back to the first figure. This is what Robin Smith says in English that Aristotle said in Ancient Greek: "... If M belongs to every N but to no X, then neither will N belong to any X. For if M belongs to no X, neither does X belong to any M; but M belonged to every N; therefore, X will belong to no N (for

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2478-463: The following valid forms of deduction for the first figure: In the Middle Ages, for mnemonic reasons they were called "Barbara", "Celarent", "Darii" and "Ferio" respectively. The difference between the first figure and the other two figures is that the syllogism of the first figure is complete while that of the second and third is not. This is important in Aristotle's theory of the syllogism for

2537-671: The formulation of logic (such as algebraic logic and Boolean logic ), and has suggested that these figures were likely to be aware of these studies in xeno-logic, and further that their acquired awareness of the shortcomings of propositional logic are likely to have stimulated their willingness to look outside the system. Indian logic attracted the attention of many Western scholars, and had an influence on pioneering 19th-century logicians such as Charles Babbage (1791-1871), Augustus De Morgan , and particularly George Boole , as confirmed by Boole's wife Mary Everest Boole in an "open letter to Dr Bose" titled "Indian Thought and Western Science in

2596-600: The fourfold negation, as evidenced by the traditions of Nagarjuna and the Madhyamaka , particularly the school of Madhyamaka given the retroactive nomenclature of Prasangika by the Tibetan Buddhist logico-epistemological tradition. Though tetralemma was also used as a form inquiry rather than logic in the Nasadiya Sukta of Rigveda (creation hymn) though seems to be rarely used as a tool of logic before Buddhism. Indian logic Indian logic stands as one of

2655-399: The hands of Bertrand Russell and A. N. Whitehead , whose Principia Mathematica (1910–13) made use of a variant of Peano's predicate logic. Term logic also survived to some extent in traditional Roman Catholic education, especially in seminaries . Medieval Catholic theology , especially the writings of Thomas Aquinas , had a powerfully Aristotelean cast, and thus term logic became

2714-461: The important aspects of Indian philosophy, logic, set theory , and especially epistemology , which Gangeśa examined rigorously, developing and improving the Nyāya scheme, and offering examples. The results, especially his analysis of cognition, were taken up and used by other darśanas. Navya-Nyāya developed a sophisticated language and conceptual scheme that allowed it to raise, analyse, and solve problems in logic and epistemology. It systematised all

2773-534: The name "two-term theory" or "term logic" – and that the reasoning process is in turn built from propositions: A proposition may be universal or particular, and it may be affirmative or negative. Traditionally, the four kinds of propositions are: This was called the fourfold scheme of propositions (see types of syllogism for an explanation of the letters A, I, E, and O in the traditional square). Aristotle's original square of opposition , however, does not lack existential import . A term (Greek ὅρος horos )

2832-691: The nature of knowledge, how knowledge is derived, and in what way knowledge can be said to be reliable. Jain logic developed and flourished from 6th century BCE to 17th century CE. According to Jains, the ultimate principle should always be logical and no principle can be devoid of logic or reason. Thus one finds in the Jain texts , deliberative exhortations on any subject in all its facts, may they be constructive or obstructive, inferential or analytical, enlightening or destructive. The Jains have doctrines of relativity used for logic and reasoning: These Jain philosophical concepts made most important contributions to

2891-418: The philosopher Gangesha Upadhyaya of Mithila . It was a development of the classical Nyāya darśana. Other influences on Navya-Nyāya were the work of earlier philosophers Vācaspati Miśra (900–980 CE) and Udayana (late 10th century). Gangeśa's book Tattvacintāmaṇi ("Thought-Jewel of Reality") was written partly in response to Śrīharśa's Khandanakhandakhādya, a defence of Advaita Vedānta, which had offered

2950-615: The premises are in the Third Figure. Symbolically, the Three Figures may be represented as follows: In Aristotelian syllogistic ( Prior Analytics , Bk I Caps 4-7), syllogisms are divided into three figures according to the position of the middle term in the two premises. The fourth figure, in which the middle term is the predicate in the major premise and the subject in the minor, was added by Aristotle's pupil Theophrastus and does not occur in Aristotle's work, although there

3009-399: The realm of applications, Boole's system could handle multi-term propositions and arguments whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's system could not deduce “No quadrangle that is a square is a rectangle that is a rhombus” from “No square that is a quadrangle is a rhombus that is a rectangle” or from “No rhombus that is

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3068-429: The realm of foundations, Boole reduced the four propositional forms of Aristotle's logic to formulas in the form of equations– by itself a revolutionary idea. Second, in the realm of logic's problems, Boole's addition of equation solving to logic– another revolutionary idea –involved Boole's doctrine that Aristotle's rules of inference (the “perfect syllogisms”) must be supplemented by rules for equation solving. Third, in

3127-609: The reason may have been that it facilitates the use of letters instead of terms avoiding the ambiguity that results in Greek when letters are used with the linking verb. In his formulation of syllogistic propositions, instead of the copula ("All/some... are/are not..."), Aristotle uses the expression, "... belongs to/does not belong to all/some..." or "... is said/is not said of all/some..." There are four different types of categorical sentences: universal affirmative (A), universal negative (E), particular affirmative (I) and particular negative (O). A method of symbolization that originated and

3186-752: The received Aristotelian logic , resulting in the observation that the Aristotelian syllogism could not account for the Indian syllogism. Max Mueller contributed an appendix to the 1853 edition of Thomson 's Outline of the Laws of Thought , in which he placed Greek and Indian logic on the same plane: "The sciences of Logic and Grammar were, as far as history allows us to judge, invented or originally conceived by two nations only, by Hindus and Greeks." Jonardon Ganeri has observed that this period saw George Boole (1815-1864) and Augustus De Morgan (1806-1871) make their pioneering applications of algebraic ideas to

3245-601: The three original traditions of logic , alongside the Greek and the Chinese logic . The Indian tradition continued to develop through early to modern times, in the form of the Navya-Nyāya school of logic. Who really knows? Who will here proclaim it? Whence was it produced? Whence is this creation? The gods came afterwards, with the creation of this universe. Who then knows whence it has arisen? The Nasadiya Sukta of

3304-491: The work of Boole (1815–1864) and Venn (1834–1923), typically yielded systems highly influenced by the term-logic tradition. The first predicate logic was that of Frege 's landmark Begriffsschrift (1879), little read before 1950, in part because of its eccentric notation. Modern predicate logic as we know it began in the 1880s with the writings of Charles Sanders Peirce , who influenced Peano (1858–1932) and even more, Ernst Schröder (1841–1902). It reached fruition in

3363-406: Was reinvigorated in the early 1970s by John Corcoran and Timothy Smiley – which informs modern translations of Prior Analytics by Robin Smith in 1989 and Gisela Striker in 2009. The Prior Analytics represents the first formal study of logic, where logic is understood as the study of arguments. An argument is a series of true or false statements which lead to a true or false conclusion. In

3422-402: Was the only way to obtain release from suffering. They therefore took great pains to identify valid sources of knowledge and to distinguish these from mere false opinions. According to the Nyaya school, there are exactly four sources of knowledge (pramanas): perception, inference, comparison and testimony. Knowledge obtained through each of these can, of course, still be either valid or invalid. As

3481-428: Was used in the Middle Ages greatly simplifies the study of the Prior Analytics. Following this tradition then, let: Categorical sentences may then be abbreviated as follows: From the viewpoint of modern logic, only a few types of sentences can be represented in this way. The fundamental assumption behind the theory is that the formal model of propositions are composed of two logical symbols called terms – hence

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