- Informal logic
- Studies the nature of natural-language arguments, including logical fallacies and paradoxes.
- Formal logic
- Systematizes modes of argumentation in terms of abstract rules.
- Symbolic logic
- Further abstraction of systems of inference using symbols for statements and logical connectives ("and", "or", and so forth).
- Major branches of symbolic logic include:
- propositional logic
- predicate logic
- Predicate logic may be further subdivided into:
- first-order logic
- second-order logic
- third-order logic
- higher-order logic
- Predicate logic may be further subdivided into:
- Mathematical logic
- Extends symbolic logic into other areas including model theory, proof theory, set theory, and recursion theory.
- Modal logic
The history of European logic is typically traced back to ancient Greece and the writings of Aristotle (384–322 BC). Logic was established as a formal discipline by Aristotle, who gave it a fundamental place in philosophy. The study of logic was part of the classical trivium, which also included grammar and rhetoric. His formulation of so-called Aristotelian logic was the dominant form of formal logic in Europe until the late 18th to early 19th centuries.
In the Eastern world, logic was developed by Buddhists and Jains, with Indian logic and Chinese logic.
In the Islamic world, Logic was further extended by Al-Farabi who categorized it into two separate groups (idea and proof). Later, Avicenna revived the study of logic and developed relationship between temporalis and the implication. Averroes defined logic as "the tool for distinguishing between the true and the false"; Richard Whately, '"the Science, as well as the Art, of reasoning"; and Frege, "the science of the most general laws of truth".
The late 18th to early 19th centuries saw the development of symbolic logic and mathematical logic introduced new paradoxes and more powerful techniques to deal with them.
History of logic
Several ancient civilizations have employed intricate systems of reasoning and asked questions about logic or propounded logical paradoxes. In Indian logic, the Nasadiya Sukta of the Rigveda (RV 10.129) contains ontological speculation in terms of various logical divisions that were later recast formally as the four circles of catuṣkoṭi: "A", "not A", "Neither A or not A", and "Both not A and not not A". The Chinese logical philosopher Gongsun Long (ca. 325–250 BC) proposed the paradox "One and one cannot become two, since neither becomes two." Also, the Chinese School of Names is recorded as having examined logical puzzles such as "A White Horse is not a Horse" as early as the fifth century BCE. In China, the tradition of scholarly investigation into logic, however, was repressed by the Qin dynasty following the legalist philosophy of Han Feizi. The earliest sustained Greek work on the subject of logic is that of Aristotle. Aristotelian logic became widely accepted in science and mathematics and remained in wide use in Europe until the early 19th century.
Islamic logic contributed to the development of modern logic, which included the development of "Avicennian logic" as an alternative to Aristotelian logic. Avicenna's system of logic was responsible for the introduction of hypothetical syllogism, temporal modal logic, and inductive logic. It later had a significant influence on European logic during the Renaissance. In Europe during the later medieval period, major efforts were made to show that Aristotle's ideas were compatible with Christian faith. During the later period of the Middle Ages, logic became a main focus of philosophers, who would engage in critical logical analyses of philosophical arguments.
In India, innovations in the scholastic school, called Nyaya, continued from ancient times into the early 18th century with the Navya-Nyaya school. By the 16th century, it 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 theory of "restrictive conditions for universals" anticipating some of the developments in modern set theory. 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 George Boole. In the 20th century, Western philosophers like Stanislaw Schayer and Klaus Glashoff have explored Indian logic more extensively.
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 symbolic logic (now called mathematical logic). In 1854, George Boole published An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, introducing symbolic logic and the principles of what is now known as Boolean logic. In 1879, Gottlob Frege published Begriffsschrift which inaugurated modern logic with the invention of quantifier notation. From 1910 to 1913, Alfred North Whitehead and Bertrand Russell published Principia Mathematica on the foundations of mathematics, attempting to derive mathematical truths from axioms and inference rules in symbolic logic. In 1931, Gödel raised serious problems with the foundationalist program and logic ceased to focus on such issues.
The development of logic since Frege, Russell and Wittgenstein had a profound influence on the practice of philosophy and the perceived nature of philosophical problems (see Analytic philosophy), and Philosophy of mathematics. Logic, especially sentential logic, is implemented in computer logic circuits and is fundamental to computer science. Logic is commonly taught by university philosophy departments often as a compulsory discipline.
- For example, Nyaya (syllogistic recursion) dates back 1900 years.
- Mohists and the school of Names date back at 2200 years.
- Averroes, In Arist. Physicam I, textus 35, ed. Juntina, IV, fol. 11vb.
- Kak, S. (2004). The Architecture of Knowledge. Delhi: CSC.
- The four Catuṣkoṭi logical divisions are formally very close to the four opposed propositions of the Greek tetralemma, which in turn are analogous to the four truth values of modern relevance logic Cf. Belnap (1977); Jayatilleke, K. N., (1967, The logic of four alternatives, in Philosophy East and West, University of Hawaii Press).
- "School of Names". Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/school-names/. Retrieved 5 September 2008.
- E.g., Kline (1972, p.53) wrote "A major achievement of Aristotle was the founding of the science of logic".
- Goodman, Lenn Evan (1992). Avicenna. Routledge. p. 184. ISBN 978-0415019293.
- Goodman, Lenn Evan (2003). Islamic Humanism. Oxford University Press. p. 155. ISBN 0195135806.
- "History of logic: Arabic logic". Encyclopædia Britannica. http://www.britannica.com/EBchecked/topic/346217/history-of-logic/65928/Arabic-logic.
- Nabavi, Lotfollah. "Sohrevardi's Theory of Decisive Necessity and kripke's QSS System". Journal of Faculty of Literature and Human Sciences. Archived from the original on 26 January 2008. http://web.archive.org/web/20080126100838/http://public.ut.ac.ir/html/fac/lit/articles.html.
- "Science and Muslim Scientists". Islam Herald. Archived from the original on 17 December 2007. http://web.archive.org/web/20071217150016/http://www.islamherald.com/asp/explore/science/science_muslim_scientists.asp.
- Hallaq, Wael B. (1993). Ibn Taymiyya Against the Greek Logicians. Oxford University Press. p. 48. ISBN 0198240430.
- Kisor Kumar Chakrabarti (June 1976), "Some Comparisons Between Frege's Logic and Navya-Nyaya Logic", Philosophy and Phenomenological Research (International Phenomenological Society) 36 (4): 554–563, http://www.jstor.org/stable/2106873, "This paper consists of three parts. The first part deals with Frege's distinction between sense and reference of proper names and a similar distinction in Navya-Nyaya logic. In the second part we have compared Frege's definition of number to the Navya-Nyaya definition of number. 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 the developments of modern set theory."
- Jonardon Ganeri (2001), Indian logic: a reader, Routledge, pp. vii, 5, 7, ISBN 0700713069
- Whitehead, Alfred North; Russell, Bertrand (1967). Principia Mathematica to *56. Cambridge University Press. ISBN 0-521-62606-4.
- Nuel Belnap, (1977). A useful four-valued logic. In Dunn & Eppstein, Modern uses of multiple-valued logic. Reidel: Boston.
- Brookshear, J. Glenn (1989). Theory of computation: formal languages, automata, and complexity. Redwood City, Calif.: Benjamin/Cummings Pub. Co.. ISBN 0805301437.
- Cohen, R.S, and Wartofsky, M.W. (1974). Logical and Epistemological Studies in Contemporary Physics. Boston Studies in the Philosophy of Science. D. Reidel Publishing Company: Dordrecht, Netherlands. ISBN 90-277-0377-9.
- Finkelstein, D. (1969). "Matter, Space, and Logic". in R.S. Cohen and M.W. Wartofsky (eds. 1974).
- Gabbay, D.M., and Guenthner, F. (eds., 2001–2005). Handbook of Philosophical Logic. 13 vols., 2nd edition. Kluwer Publishers: Dordrecht.
- Hilbert, D., and Ackermann, W, (1928). Grundzüge der theoretischen Logik (Principles of Mathematical Logic). Springer-Verlag. OCLC 2085765
- Susan Haack, (1996). Deviant Logic, Fuzzy Logic: Beyond the Formalism, University of Chicago Press.
- Hodges, W., (2001). Logic. An introduction to Elementary Logic, Penguin Books.
- Hofweber, T., (2004), Logic and Ontology. Stanford Encyclopedia of Philosophy. Edward N. Zalta (ed.).
- Hughes, R.I.G., (1993, ed.). A Philosophical Companion to First-Order Logic. Hackett Publishing.
- Kline, Morris (1972). Mathematical Thought From Ancient to Modern Times. Oxford University Press. ISBN 0-19-506135-7.
- Kneale, William, and Kneale, Martha, (1962). The Development of Logic. Oxford University Press, London, UK.
- Liddell, Henry George; Scott, Robert. "Logikos". A Greek-English Lexicon. Perseus Project. http://www.perseus.tufts.edu/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3D%2363716. Retrieved 8 May 2009.
- Mendelson, Elliott, (1964). Introduction to Mathematical Logic. Wadsworth & Brooks/Cole Advanced Books & Software: Monterey, Calif. OCLC 13580200
- Harper, Robert (2001). "Logic". Online Etymology Dictionary. http://www.etymonline.com/index.php?term=logic. Retrieved 8 May 2009.
- Smith, B., (1989). "Logic and the Sachverhalt". The Monist 72(1):52–69.
- Whitehead, Alfred North and Bertrand Russell, (1910). Principia Mathematica. Cambridge University Press: Cambridge, England. OCLC 1041146
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- LogicWiki, an external wiki
- Introductions and tutorials
- An Introduction to Philosophical Logic, by Paul Newall, aimed at beginners.
- forall x: an introduction to formal logic, by P.D. Magnus, covers sentential and quantified logic.
- Logic Self-Taught: A Workbook (originally prepared for on-line logic instruction).
- Nicolas Rescher. (1964). Introduction to Logic, St. Martin's Press.
- "Symbolic Logic" and "The Game of Logic", Lewis Carroll, 1896.
- Math & Logic: The history of formal mathematical, logical, linguistic and methodological ideas. In The Dictionary of the History of Ideas.
- Reference material
- Translation Tips, by Peter Suber, for translating from English into logical notation.
- Ontology and History of Logic in Western Thought. An Introduction Annotated bibliography
- Reading lists