History of logic
The history of logic documents the development of logic as it occurs in various 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 B.C.E.. The notions of systems of reasoning and logic, however, are sufficiently imprecise that various answers to the questions of what they are and how they are to be understood have been given. 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 logicians in Medieval Europe.
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
Two of the six Indian schools of thought deal with logic: Nyaya and Vaisheshika. 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-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, two main competing logical traditions emerged. Stoic logic traced its roots back to Euclid of Megara (c. 430 - c. 360 B.C.E.), a pupil of Socrates, and with its concentration on propositional logic was perhaps closer to modern logic. The Megarians were interested in puzzles, and studied modality and conditionals. The Stoics used numbers as variables for replacing whold propositions. The most important Stoic logician was Chrysippus (c. 279 - 206 B.C.E.), who discussed five basic or valid inference schemata, and from them derived or proved many other valid inference schemata.
There was also a medieval tradition that held that the Greek philosopher Parmenides (5th century B.C.E.) invented logic while living on a rock in Egypt. In any case, his disciple, Zeno of Elea (5th century B.C.E.) did produce many supposedly logical arguments, known as Zeno's paradoxes. These were given in support of Parmenides' philosophy—a philosophy that denied motion and multiplicity—and purported to show that a non-Parmenidean view leads to absurdity. This method of proving something by assuming its alternative and showing that this assumption leads to absurdity is known as reduction ad absurdum and Zeno's use of it suggests that he knew of the general pattern of such argument. Zeno's paradoxes do, however, all contain fatal mistakes, but showing what the mistakes are often required waiting until much later developments in logic and mathematical logic.
The Greek tradition that survived to influence later cultures, however, was the Peripatetic tradition which originated in Aristotle's collection of works known as the "Organon" or instrument, the first systematic Greek work on logic. In fact, Aristotle is often called the first great logician. Although he did not use these terms himself, Aristotle introduced the formal study of what is now known as formal logic, that is logic that is concerned with the form, not the content, of statements or propositions, and the relationships that exist between different statements on the basis of their form—some statements being accepted (as premises), other statement(s) follow (as conclusion(s)) from those accepted statements because of their form.
Aristotle held that a proposition involves two terms, a subject and a predicate. Propositions can be universal ("all," "no") or particular ("some"), and affirmative or negative. Aristotle's formal logic was confined to examination of syllogisms, which consist of three propositions. The first two are the premises, and must share only one term. The third propositon is the conclusion, which contains the two terms that are not shared by the premises. Aristotle also investigated how the common term (shared by the two premises) can occur and the effects of its different ways of occurring. Aristotle's work work on syllogisms bears interesting comparison with the Indian schema of inference and the less rigid Chinese discussion.
Aristotle also formulated certain theses about logic (sometimes called metalogical principles): The Law of Noncontradiction, the Principle of the Excluded Middle, and the Law of Bivalence. In addition he investigated some of what are now known as informal fallacies, fallacies that occur for some reason other than the form of the argument, such as argumetum ad hominem, and appeal to the crowd.
Aristotle's successor as head of his school, Theophrastus of Eresus (c. 371 - c. 286 B.C.E.), carried on Aristotle's investigations of logic and added to them.
Through Latin in Western Europe, and disparate languages more to the East, such as Arabic, Armenian and Georgian, the Aristotelian tradition was considered to codify pre-eminently the laws of reasoning. It was only in the Nineteenth Century that this viewpoint changed; it has been suggested by a few commentators that this change may have been facilitated by an acquaintance with the classical literature of India and deeper knowledge of China.
Except for what was done in the Arabic world, there was little work in logic between that of Boethius (480 - 524 or 525 C.E.) and Peter Abelard (1079-1142) in the 12th century.
Logic in Islamic philosophy
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 was 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
Medieval Logic (also known as Scholastic Logic) generally means the form of Aristotelian logic developed in medieval Occident throughout the period c. 1200-1600. The first
During this period mnemonic names were created for the valid moods of the syllogism that had been discussed in Aristotle's Prior Analytics. Two of those moods were BARBARA, in which the three propositions of the syllogism consist entirely of universal affirmatives, and CELARENT, in which one premise is a universal negative, the other a universal affirmative, and the conclusion is a universal negative.
Logic in the Medieval Period was developed through textbooks such as that by Peter of Spain (fl. thirteenth century, but whose exact identity is unknown), who was the author of a standard textbook on logic, the Tractatus which was well known in Europe for many centuries. This tradition of Medieval logic reached its high point in the fourteenth century, with the works of William of Ockham (c. 1287-1347) and Jean Buridan.
One feature was the Development of Aristotelian logic through what is known as Supposition Theory, a study of the semantics of the terms of the proposition. The last great works in this tradition are the Logic of John Poinsot (1589-1644, known as John of St Thomas), and the Metaphysical Disputations of Francisco Suarez (1548-1617).
In the sixteenth century, however, what we now know as logic was largely displaced by interest in and study of dialectic. Thus the three works of Philip Melanchthon (1497-1560), Compendiaria dialectics ratio (1520), Dialectics libri quattuor (1528), and Erotemata dialectics (1547) each carried the term dialectics in its title, instead of logic, and the same was true of works by Petrus Ramus (1515-1572) and the scholar known as the Portugese Aristotle, Petrus Fonseca, S.J., whose Institutionum dialecticarum libri octo first appeared in 1564.
In the eighteenth century, there was a return to the use of logic. Christoph Scheibler (1589-1653), known as the Protestant Suarez, published an encyclopedic book Opus Logicum in Marburg, Germany, in 1633. Other books with the term logic in their titles appeared, such as Logica Hamburgensis in 1638 from Joachim Jungius (1587-1657), Logica vetus et nova (1654) by the German Cartesian Johannes Clauberg (1622-1655), and some others. The most notable and important work of this era was the Port Royal Logic.
Traditional Logic
What has become known as traditional logic generally means the textbook tradition that begins with Antoine Arnauld and Pierre Nicole's Logic, or the Art of Thinking, better known as the Port-Royal Logic. Published in 1662, it was the most influential work on logic in England until Mill's System of Logic in 1825. The book presents a loosely Cartesian doctrine (that the proposition is a combining of ideas rather than terms, for example) within a framework that is broadly derived from Aristotelian and medieval term logic. Between 1664 and 1700 there were eight editions, and the book had considerable influence after that. It was frequently reprinted in English up to the end of the nineteenth century.
The account of propositions that Locke gives in the Essay is essentially that of Port-Royal. "Verbal propositions, which are words, [are] the signs of our ideas, put together or separated in affirmative or negative sentences. So that proposition consists in the putting together or separating these signs, according as the things which they stand for agree or disagree." (Locke, An Essay Concerning Human Understanding, IV. 5. 6)
Works in this tradition include Isaac Watts' Logick: Or, the Right Use of Reason (1725), Richard Whately's Logic (1826), and John Stuart Mill's A System of Logic (1843), which was one of the last great works in the tradition.
Modern Logic
The idea of a calculus of reasoning was cultivated by Gottfried Wilhelm Leibniz, who was the first to formulate the notion of a broadly applicable system of mathematical logic. However, the relevant documents were not published until 1901 or remain unpublished to the present day, and the current understanding of the power of Leibniz's discoveries did not emerge until the 1980s. [See Lenzen's chapter in Gabbay and Woods (2004)].
John Venn 1834-1923, was a Cambridge logician who published three standard texts in logic, The Logic of Chance 1866, Symbolic Logic 1881, and The Principles of Empirical Logic 1889. Today he is remembered mostly for his logical diagrams, known as Venn diagrams, used for representing syllogisms. He was not the originator of using geometrical representations to illustrate syllogistic logic; Leibniz had often used such methods. Venn became critical of the diagrams used in the nineteenth century, especially those of logicians George Boole 1815-1864, and Augustus de Morgan 1806-1871. Boole was the inventor of what is now known as Boolean algebra, which is the basis of all modern computer arithmetic; he is regarded as being one of the founders of the field of computer science, although computers did not exist in his day. De Morgan was an Indian-born British mathematician and logician who formulated what are now known as De Morgan's laws and was the first to introduce the term mathematical induction and make rigorous the idea. Venn wrote the book Symbolic Logic to interpret and make his corrections on Boole's work. Prior to publishing this book, Venn wrote a paper entitled "On the Diagrammatic and Mechanical Representation of Prepositions and Reasonings" introducing Venn diagrams. This paper was published in the Philosophical Magazine and Journal of Science in July, 1880. In Symbollic Logic, Venn further elaborated on these diagrams, and they became the most important part of his work.
In an 1885 article read by Giuseppe Peano, Ernst Schröder, and others, Charles Peirce introduced the term second-order logic and provided us with much of our modern logical notation, including prefixed symbols for universal and existential quantification. Logicians in the late 19th and early 20th centuries were thus more familiar with the Peirce-Schröder system of logic, although Frege is generally recognized today as being the "Father of modern logic."
Gottlob Frege has been called the greatest logician since Aristotle. His work was the foundation or beginning point for an enormous outpouring of work in formal logic, beginning at the end of the 19th and continuing into the 20th century. Frege's 1879 Begriffsschrift extended formal logic beyond propositional logic to include constructors such as "all," and "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." Frege's peculiar two dimensional notation led to his work being ignored for many years.
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
- Term Logic
- Ernst Schröder
- Charles Peirce
ReferencesISBN links support NWE through referral fees
- Church, Alonzo, "A Bibliography of Symbolic Logic," Journal of Symbolic Logic 1: 121-218; 3:178-212, 1936-38.
- Dumitriu, Anton, History of Logic, Volume III, Revised and updated and enlarged translation of the Roumanian work, Tunbridge Wells, Kent, UK: Abacus Press, 1977. ISBN 0856261424
- Gabbay, Dov, and Woods, John, eds, Handbook of the History of Logic. Vol. 1: Greek, Indian and Arabic logic; Vol. 3: The Rise of Modern Logic I: Leibniz to Frege, Amsterdam & Boston: Elsevier, 2004. ISBN 0444515968 (set)
- Grattan-Guinness, Ivor, The Search for Mathematical Roots 1870-1940, Princeton NJ: Princeton University Press, 2000. ISBN 069105858X ISBN 9780691058580
- Haack, Susan, Deviant Logic, Fuzzy Logic, Chicago & London: University of Chicago Press, 1996. ISBN 0226311333 (cloth) ISBN 0226311341 (paper)
- Kneale, William and Martha, The Development of Logic. New York: Oxford University Press, 1962, 1985. ISBN 0198247737 ISBN 9780198247739
- Putnam, Hilary, Philosophy of Logic, New York, Harper & Row, 1971. ISBN 0061360422 (pbk)
- Quine, W.V., Philosophy of Logic, Cambridge, Mass.: Harvard University Press, 1986. ISBN 0674665635
- Van Heijenoort, Jean, Selected Essays: History of Logic 3, Napoli: Bibliopolis, 1985. ISBN 8870881229
External links
- Annotated bibliography on the history of logic
- Overview of Indian and Greek Development of Logic and Language
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