Practicing logicians and philosophers, on the other hand, have been very into the limits of reason for decades.
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On a more general level, we need to get more people working on the problem, including mathematicians, logicians, ethicists, economists, social scientists, and philosophers.
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The president had created what logicians call a false dilemma, that support for free speech or for teammates equated to disrespect for flag, anthem or country.
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The debate over what mix of techniques to use is obscured by what logicians call the fallacy of composition – the failure to recognise that the whole can be quite different from the sum of the parts.
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For the form of contrarieties is multiplex, as logicians teach.
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Wael Hallaq (1993), Ibn Taymiyya Against the Greek Logicians, p. 48. Oxford University Press, .
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According to most logicians, the three primary mental operations are apprehension (understanding), judgement, and inference.
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Logicians have many different views on the nature of material implication and approaches to explain its sense.
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12 (2006), pp. 219–40. namely, 'letter'; but every time that word type is written, a new word token has been created. Some logicians consider a word type to be the class of its tokens. Other logicians counter that the word type has a permanence and constancy not found in the class of its tokens.
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Polish Logic is an anthology of papers by several authors—Kazimierz Ajdukiewicz, Leon Chwistek, Stanislaw Jaskowski, Zbigniew Jordan, Tadeusz Kotarbinski, Stanisław Leśniewski, Jan Łukasiewicz, Jerzy Słupecki, and Mordchaj Wajsberg—published in 1967 and covering the period 1920-1939\. The work focuses on the contributions of Polish logicians, more particularly, mathematical logicians, to modern logic.
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In 2012 an international conference on Vasiliev's work was held in Moscow where a number of important modern paraconsistent logicians contributed.
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These logicians use the phrase not p or q for the conditional connective and the term implies for an asserted implication relation.
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In the investigation of the semantics of language, many logicians find themselves facing this paradox. An example of application can be seen in the inherent concern of logicians with the conditions of truth within a sentence, and not, in fact, with the conditions under which a sentence can be truly asserted. Additionally, the paradox has been shown to have applications in industry.
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He also rejected argument from analogy. His doctrinal heirs, the Stoic logicians, inaugurated the most important school of logic in antiquity other than Aristotle's peripatetics.
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Skerpan-Wheeler, Elizabeth. "John Milton." British Rhetoricians and Logicians, 1500–1660: Second Series. Ed. Edward A. Malone. Detroit: Gale, 2003. Dictionary of Literary Biography Vol. 281.
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Monimus of Syracuse Monimus (; ; 4th century BC) of Syracuse, was a Cynic philosopher who endorsed philosophical skepticism, denying that there was a criterion of truth.Sextus Empiricus, Against the Logicians, 7.88.
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In the history of mathematics, Alfred Tarski (1901–1983) is one of the most important logicians. His name is now associated with a number of theorems and concepts in that field.
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He was the Maximum Leader for the Logicians Liberation League, and drafted its manifesto.Logicians Liberation League Manifesto Meyer died of lung cancer on 6 May 2009, aged 76, in Canberra, Australia.
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The calculus of individuals is the starting point for the post-1970 revival of mereology among logicians, ontologists, and computer scientists, a revival well-surveyed in Simons (1987) and Casati and Varzi (1999).
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One of the schools of Mohism that has received some attention is the Logicians school, which was interested in resolving logical puzzles. Not much survives from the writings of this school, since problems of logic were deemed trivial by most subsequent Chinese philosophers. Historians such as Joseph Needham have seen this group as developing a precursor philosophy of science that was never fully developed, but others believe that recognizing the Logicians as proto-scientists reveals too much of a modern bias.
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He also made use of inductive logic, such as the methods of agreement, difference, and concomitant variation which are critical to the scientific method.Goodman, Lenn Evan (2003), Islamic Humanism, p. 155, Oxford University Press, . One of Avicenna's ideas had a particularly important influence on Western logicians such as William of Ockham: Avicenna's word for a meaning or notion (ma'na), was translated by the scholastic logicians as the Latin intentio; in medieval logic and epistemology, this is a sign in the mind that naturally represents a thing.
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Zinoviev played an important role in the development of national logic in the 1950–1960s. His early program of "substantive logic" did not receive official recognition, but influenced the development of Soviet research on the methodology of science. In the 1960s, Zinoviev was one of the leading Soviet logicians, the leader of the "cognitive movement", which, according to Vladislav Lektorsky, fascinated many philosophers, logicians, mathematicians, psychologists and linguists. Five works by Zinoviev were published in the West, which was a unique case for Russian philosophical thought.
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Gongsun Long (, BCZhou, Yunzhi, "Gongsun Long". Encyclopedia of China (Philosophy Edition), 1st ed.Liu 2004, p. 336) was a Chinese philosopher and writer who was a member of the School of Names (Logicians) of ancient Chinese philosophy.
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The Australasian Association for Logic (AAL) is a philosophical organisation for logicians in Australia and New Zealand. The purpose of the organisation is to promote the study of logic. The association publishes the Australasian Journal of Logic.
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This drove him to bankruptcy and eventually suicide.Sextus Empiricus, Against the Logicians 1.107-8; a much later work by a philosopher, not a historian. The work may have been completed by Laches, also an inhabitant of Lindos.
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The date chosen to celebrate World Logic Day, 14 January, corresponds to the date of death of Kurt Gödel and the date of birth of Alfred Tarski, two of the most prominent logicians of the twentieth century.
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Other Islamic scholars at the time, however, argued that the term Qiyas refers to both analogical reasoning and categorical syllogism in a real sense.Wael B. Hallaq (1993), Ibn Taymiyya Against the Greek Logicians, p. 48. Oxford University Press, .
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Some logicians favor a multiple-conclusion consequence relation over the more traditional single-conclusion relation on the grounds that the latter is asymmetric (in the informal, non-mathematical sense) and favors truth over falsity (or assertion over denial).
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Other Islamic scholars at the time, however, argued that the term Qiyas refers to both analogical reasoning and categorical syllogism in a real sense.Wael B. Hallaq (1993), Ibn Taymiyya Against the Greek Logicians, p. 48. Oxford University Press, .
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The first logicians to debate conditional statements were Diodorus Cronus and his pupil Philo. Writing five-hundred years later, Sextus Empiricus refers to a debate between Diodorus and Philo.Sextus Empiricus, Pyr. Hyp. ii. 110–112; Adv. Math. viii.
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David de Rodon or plain Derodon (c. 1600 - 1664), was a French Calvinist theologian and philosopher. Derodon was born at Die, in the Dauphiné. He had the reputation of being one of the most eminent logicians of his time.
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Hare, Richard M. (1967). Some alleged differences between imperatives and indicatives. Mind, 76, 309-326. There is no consensus among logicians about the truth or falsity of these (or similar) claims and mixed imperative and declarative inference remains vexed.
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Eves observes "that logicians have endeavored to push down further the starting level of the definitional development of mathematics and to derive the theory of sets, or classes, from a foundation in the logic of propositions and propositional functions". But by the late 19th century the logicians' research into the foundations of mathematics was undergoing a major split. The direction of the first group, the Logicists, can probably be summed up best by – "to fulfil two objects, first, to show that all mathematics follows from symbolic logic, and secondly to discover, as far as possible, what are the principles of symbolic logic itself." The second group of logicians, the set-theorists, emerged with Georg Cantor's "set theory" (1870–1890) but were driven forward partly as a result of Russell's discovery of a paradox that could be derived from Frege's conception of "function", but also as a reaction against Russell's proposed solution.
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Logic should consist of open, not covert, reasoning. It should be simple and unmixed, with no hidden inferences. Previous logicians incorrectly considered all four figures as being simple and pure. The four figures were created by playfully changing the middle term’s position.
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Modern logicians disagree concerning the nature of argument constituents. Quine devotes the first chapter of Philosophy of Logic to this issue.Willard Quine, Philosophy of logic, Harvard, 1970/1986. Historians have not even been able to agree on what Aristotle took as constituents.
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He would call for a maieutic (jurisprudential) oratory art against the grain of the modern privilege of the dogmatic form of reason, in what he called the "geometrical method" of René Descartes and the logicians at the Port-Royal- des-Champs abbey.
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Later ancient logicians (including Aristotle) and practitioners of other ancient sciences have employed diairetic modes of classification, e.g., to classify plants in ancient biology. Although classification is still an important part of science, diairesis has been abandoned and is now of historical interest only.
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Introduction to Buridan: Sophisms on Meaning and Truth, Appleton-Century-Crofts The Medieval logicians give elaborate sets of syntactical rules for determining when a term supposits discretely, determinately, confusedly, or confusedly and distributively. So for example the subject of a negative claim, or indefinite one supposits determinately, but the subject of a singular claim supposits discretely, while the subject of an affirmative claim supposits confusedly and determinately. Albert of Saxony gives 15 rules for determining which type of personal supposition a term is using. Further the medieval logicians did not seem to dispute about the details of the syntactic rules for determining type of personal supposition.
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Muzaffar Iqbal, Science and Islam, pg. 120. From the Greenwood Guides to Science and Religion Series. Westport: Greenwood Publishing Group, 2007. According to the Routledge Encyclopedia of Philosophy: Important developments made by Muslim logicians included the development of "Avicennian logic" as a replacement of Aristotelian logic.
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Mally's metaphysical work influences some contemporary metaphysicians and logicians working in abstract object theory, especially Edward Zalta. The analytic philosopher David Kellogg Lewis argued forcefully that the name of the fictional Australian poet Ern Malley, created by James McAuley and Harold Stewart, was an allusion to Mally.
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The School of Names (), sometimes called the School of Forms and Names (), was a school of Chinese philosophy that grew out of Mohism during the Warring States period in 479–221 BCE. The followers of the School of Names were sometimes called the Logicians or Disputers.
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Prior to the mid-12th century, medieval logicians were only familiar with a portion of Aristotle's works, including such titles as Categories and On Interpretation, works that contributed heavily to the prevailing Old Logic, or logica vetus. The onset of a New Logic, or logica nova, arose alongside the reappearance of Prior Analytics, the work in which Aristotle developed his theory of the syllogism. Prior Analytics, upon re- discovery, was instantly regarded by logicians as "a closed and complete body of doctrine," leaving very little for thinkers of the day to debate and reorganize. Aristotle's theory on the syllogism for assertoric sentences was considered especially remarkable, with only small systematic changes occurring to the concept over time.
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See the next section. Similar views, perhaps not similarly motivated, are found in later logicians, including Gottlob Frege (1848–1925). Some recent formulations of standard one-sorted first-order logic seem to be in accord with a form of it, if they do not actually imply the principle itself.Corcoran, John.
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Some logicians, such as Paul Grice, have used conversational implicature to argue that, despite apparent difficulties, the material conditional is just fine as a translation for the natural language 'if...then...'. Others still have turned to relevance logic to supply a connection between the antecedent and consequent of provable conditionals.
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Even using ordinary set theory and binary logic to reason something out, logicians have discovered that it is possible to generate statements which are logically speaking not completely true or imply a paradox,Patrick Hughes & George Brecht, Vicious Circles and Infinity. An anthology of Paradoxes. Penguin Books, 1978. Nicholas Rescher, Epistemological Studies.
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The Nine Schools of Thought which came to dominate the others were Confucianism (as interpreted by Mencius and others), Legalism, Taoism, Mohism, the utopian communalist Agriculturalism, two strains of Diplomatists, the sophistic Logicians, Sun-tzu's Militarists, and the Naturalists..Carr, Brian & al. Companion Encyclopaedia of Asian Philosophy, p. 466. Taylor & Francis, 2012. , 9780415035354.
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Parsons works on the semantics of natural language to develop theories of truth and meaning for natural language similar to those devised for artificial languages by philosophical logicians. Heavily influenced by Alexius Meinong, he wrote Nonexistent Objects (1980), which dealt with possible world theory in order to defend the reality of nonexistent objects.
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We think for ourselves and draw our own conclusions; knowledge permits freedom to act. Yet the way inferences are drawn is subject to the influence of logicians, such as Aristotle, and their language.Inference and Persuasion, pp. 1–22 Aristotle's focus on class inclusion and exclusion highlights the limits of the language of logic.
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Another theorem of his concerns the constructible sets in algebraic geometry, i.e. those in the Boolean algebra generated by the Zariski-open and Zariski-closed sets. It states that the image of such a set by a morphism of algebraic varieties is of the same type. Logicians call this an elimination of quantifiers.
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According to Sextus Empiricus, Anaxarchus "compared existing things to a scene-painting and supposed them to resemble the impressions experienced in sleep or madness."Sextus Empiricus, Against the Logicians, 7.88. Anaxarchus's student Pyrrho is said to have adopted "a most noble philosophy, … taking the form of agnosticism and suspension of judgement."Diogenes Laërtius, Lives, ix.
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2), and Lucas (2000: chpt. 10). The entry Whitehead's point-free geometry includes two contemporary treatments of Whitehead's theories, due to Giangiacomo Gerla, each different from the theory set out in the next section. Although mereotopology is a mathematical theory, we owe its subsequent development to logicians and theoretical computer scientists. Lucas (2000: chpt.
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Rotation in space is achieved by use of quaternions, and Lorentz transformations of spacetime by use of biquaternions. Early in the 20th century, hypercomplex number systems were examined. Later their automorphism groups led to exceptional groups such as G2. In the 1890s logicians were reducing the primitive notions of synthetic geometry to an absolute minimum.
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Languages use different strategies for expressing counterfactuality. Languages may have a dedicated counterfactual morphemes, or they may recruit some combination of tense, aspect, and mood morphemes. Since the early 2000s, linguists, philosophers of language, and philosophical logicians have intensely studied the nature of this grammatical marking, and it continues to be an active area of study.
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Leibniz has been noted as one of the most important logicians between the times of Aristotle and Gottlob Frege.Lenzen, W., 2004, "Leibniz's Logic," in Handbook of the History of Logic by D. M. Gabbay/J. Woods (eds.), volume 3: The Rise of Modern Logic: From Leibniz to Frege, Amsterdam et al.: Elsevier-North-Holland, pp. 1–83.
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Kant summed up his thoughts on this topic in a short footnote that appeared in the second edition of the Critique of Pure Reason, B141. He had been discussing the definition of judgment in general. Logicians had usually defined it as a relation between two concepts. Kant disagreed because, he claimed, only categorical judgments are so defined.
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Argument identification is the identification of arguments in a text or spoken discourse. Many or most of the statements will not be arguments or parts of arguments. But some of those statements might look similar to arguments. Informal logicians have especially noted the similarity between words used to express arguments and those used to express explanations.
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While the material conditional operator used in classical logic is sometimes read aloud in the form of a conditional sentence, the intuitive interpretation of conditional statements in natural language does not always correspond to it. Thus, philosophical logicians and formal semanticists have developed a wide variety of conditional logics which better match actual conditional sentences and actual conditional reasoning.
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Wuji references are found in Chinese classic texts associated with diverse schools of Chinese philosophy, including Taoism, Confucianism, and School of Names. Zhang and Ryden summarize the philosophical transformation of wuji "limitless". > The expression 'limitless' and its relatives are found in the Laozi and the > Zhuangzi and also in writings of the logicians. It has no special > philosophical meaning.
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Logicians in the western traditions have often expressed belief in some other logical quality besides affirmation and denial. Sextus Empiricus, in the 2nd or 3rd century CE, argued for the existence of "nonassertive" statements, which indicate suspension of judgment by refusing to affirm or deny anything.Sextus Empiricus, Outlines of Pyrrhonism. R.G. Bury (trans.) (Buffalo: Prometheus Books, 1990).
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Therefore, take Foundations of Arithmetic off the table! Notice that this argument is composed of both imperatives and declaratives and has an imperative conclusion. Mixed inferences are of special interest to logicians. For instance, Henri Poincaré held that no imperative conclusion can be validly drawn from a set of premises which does not contain at least one imperative.
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For example, "God loves humanity", really means "God is a lover of humanity", "God exists" means "God is a thing". This theory of judgment dominated logic for centuries, but it has some obvious difficulties: it only considers proposition of the form "All A are B.", a form logicians call universal. It does not allow propositions of the form "Some A are B", a form logicians call existential. If neither A nor B includes the idea of existence, then "some A are B" simply adjoins A to B. Conversely, if A or B do include the idea of existence in the way that "triangle" contains the idea "three angles equal to two right angles", then "A exists" is automatically true, and we have an ontological proof of A's existence.
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A number of solutions have been put forward. Careful analyses have been made by some logicians. Though solutions differ, they all pinpoint semantic issues concerned with counterfactual reasoning. We want to compare the amount that we would gain by switching if we would gain by switching, with the amount we would lose by switching if we would indeed lose by switching.
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Rasiowa became strongly influenced by Polish logicians. She wrote her Master's thesis under the supervision of Jan Łukasiewicz and Bolesław Sobociński. In 1944, the Warsaw Uprising broke out and consequently Warsaw was almost completely destroyed. This was not only due to the immediate fighting, but also because of the systematic destruction which followed the uprising after it had been suppressed.
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Raymond Merrill Smullyan (; May 25, 1919 – February 6, 2017) was an American mathematician, magician, concert pianist, logician, Taoist, and philosopher. Born in Far Rockaway, New York, his first career was stage magic. He earned a BSc from the University of Chicago in 1955 and his Ph.D. from Princeton University in 1959. He is one of many logicians to have studied with Alonzo Church.
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Although many practical instances of SAT can be solved by heuristic methods, the question of whether there is a deterministic polynomial-time algorithm for SAT (and consequently all other NP-complete problems) is still a famous unsolved problem, despite decades of intense effort by complexity theorists, mathematical logicians, and others. For more details, see the article P versus NP problem.
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Following the work of logicians like Godehard Link and linguists like Manfred Krifka, we know that the mass/count distinction can be given a precise mathematical definition in terms of notions like cumulativity and quantization. Discussed by Barry Schein in 1993, a new logical framework, called plural logic, has also been used for characterizing the semantics of count nouns and mass nouns.
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It founded what was first known as the "algebra of logic" tradition.Witold Marciszewski (editor), Dictionary of Logic as Applied in the Study of Language (1981), pp. 194–5. Among his many innovations is his principle of wholistic reference, which was later, and probably independently, adopted by Gottlob Frege and by logicians who subscribe to standard first-order logic. A 2003 articleCorcoran, John (2003).
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The laws are named after Augustus De Morgan (1806–1871),DeMorgan’s Theorems at mtsu.edu who introduced a formal version of the laws to classical propositional logic. De Morgan's formulation was influenced by algebraization of logic undertaken by George Boole, which later cemented De Morgan's claim to the find. Nevertheless, a similar observation was made by Aristotle, and was known to Greek and Medieval logicians.
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Mozi responded, "Chu territory is two thousand kilometers square, (while) Song territory is (only) two hundred kilometers square: these are like the decorated carriage and the dilapidated cart!", which convinced the king to cancel the attack. School of Names or Logicians focused upon the relationship between words and reality. The Shizi links the Logician doctrine of fen "separation; distribution; allocation" with the Confucian rectification of names.
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In mathematics and computer science, the gradations of applicable meaning of a fuzzy concept are described in terms of quantitative relationships defined by logical operators. Such an approach is sometimes called "degree-theoretic semantics" by logicians and philosophers,Roy T. Cook, A dictionary of philosophical logic. Edinburgh University Press, 2009, p. 84. but the more usual term is fuzzy logic or many-valued logic.
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Some philosophers and logicians disagree with the philosophical conclusions that Chaitin has drawn from his theorems related to what Chaitin thinks is a kind of fundamental arithmetic randomness.Panu Raatikainen, "Exploring Randomness and The Unknowable" Notices of the American Mathematical Society Book Review October 2001. The logician Torkel Franzén criticized Chaitin's interpretation of Gödel's incompleteness theorem and the alleged explanation for it that Chaitin's work represents.
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Gottfried Wilhelm (von) Leibniz (;"Leibniz" entry in Collins English Dictionary. or ; – 14 November 1716) was a prominent German polymath and one of the most important logicians, mathematicians and natural philosophers of the Enlightenment. As a representative of the seventeenth-century tradition of rationalism, Leibniz developed, as his most prominent accomplishment, the ideas of differential and integral calculus, independently of Isaac Newton's contemporaneous developments. Extract of page 469.
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Further on, he explored the theme of the interval in his On Angels, a book of angelology, arising at the intersection between metaphysics and the philosophy of religion. In this book, the entities called angels are analysed as a cases of "beings of the interval". The entire exercise has thus the secondary value of ontology of the interval. Sorin Vieru is one of the few Păltiniş logicians.
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An immediate mark is positioned between a subject and a remote mark (predicate). in the premises of a pure syllogism. The only reason that this was generally accepted, Kant remarked, was that the logicians had made people believe that all of the other kinds of judgments could be reduced to being categorical judgments. Kant claimed to have disproved this in his Critique, A 73.
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Buno Ramnath, Shankara Tarkabagish and other scholars and logicians made the name of Nabadwip famous in the eighteenth century. Shakti worship spread in Nabadwip during the time of Raja Krishnachandra of Nadia royal family. During the reign of Raja Krishnachandra Roy and later Raja Girish Chandra, the popularity, glory and pomp of Shakta Rash increased. Various temples and idols were established in that time.
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Other writers were famous Dvaita saints of the Udupi order such as Jayatirtha (earning the title Tikacharya for his polemicial writings), Vyasatirtha who wrote rebuttals to the Advaita philosophy and of the conclusions of earlier logicians, and Vadirajatirtha and Sripadaraya both of whom criticized the beliefs of Adi Sankara. Apart from these saints, noted Sanskrit scholars adorned the courts of the Vijayanagara kings and their feudal chiefs.
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These two aspects of theory and practice have been developed almost entirely independently of each other. The main reason is undoubtedly that logicians are interested in questions radically different from those with which the applied mathematicians and electrical engineers are primarily concerned. It cannot, however, fail to strike one as rather strange that often the same concepts are expressed by very different terms in the two developments.
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Towards the end of antiquity Stoic logic was neglected in favour of Aristotle's logic, and as a result the Stoic writings on logic did not survive, and the only accounts of it were incomplete reports by other writers. Knowledge about Stoic logic as a system was lost until the 20th-century when logicians familiar with the modern propositional calculus reappraised the ancient accounts of it.
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There are several alternative theories of the cognitive processes that human reasoning is based on.Byrne, R.M.J. and Johnson-Laird, P.N. (2009).'If' and the problems of conditional reasoning. Trends in Cognitive Sciences, 13, 282-287 One view is that people rely on a mental logic consisting of formal (abstract or syntactic) inference rules similar to those developed by logicians in the propositional calculus.O’Brien, D. (2009).
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A similar situation arises with 2 + 2 = 5, which is necessarily false: : If 2 + 2 = 5, then Bill Gates graduated in Medicine. Some logicians view this situation as indicating that the strict conditional is still unsatisfactory. Others have noted that the strict conditional cannot adequately express counterfactual conditionals,Jens S. Allwood, Lars-Gunnar Andersson, and Östen Dahl, Logic in Linguistics, Cambridge University Press, 1977, , p. 120.
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Becker published his major work, Mathematical Existence in the Yearbook in 1927, the same year Martin Heidegger's Being and Time appeared there. Becker attended Heidegger's seminars at this period. Becker utilized not only Husserlian phenomenology but, much more controversially, Heideggerian hermeneutics, discussing arithmetical counting as "being toward death". His work was criticized both by neo-Kantians and by more mainstream, rationalist logicians, to whom Becker feistily replied.
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Kant holds that the definition of truth is merely nominal and, therefore, we cannot employ it to establish which judgements are true. According to Kant, the ancient skeptics were critical of the logicians for holding that, by means of a merely nominal definition of truth, they can establish which judgements are true. They were trying to do something that is "impossible without qualification and for every man".
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Semanticists and logicians sometimes draw a distinction between coreference and what is known as a bound variable.For discussions of bound variables, see for instance Portner (2005:102ff.). An instance of a bound variable can look like coreference, but from a technical standpoint, one can argue that it actually is not. Bound variables occur when the antecedent to the proform is an indefinite quantified expression, e.g.
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His contributions to semantics, especially to the maturing theory of supposition, are still studied by logicians. William of Ockham was probably the first logician to treat empty terms in Aristotelian syllogistic effectively; he devised an empty term semantics that exactly fit the syllogistic. Specifically, an argument is valid according to William's semantics if and only if it is valid according to Prior Analytics.John Corcoran (1981).
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Classics in the History of Psychology. Eprint. and generally in philosophy by Maurice Merleau-Ponty who held the chairs of philosophy and child psychologyReynolds, Jack (as last updated 2005), "Maurice Merleau-Ponty (1908–1961)", Internet Encyclopedia of Philosophy. Eprint. at the University of Paris. Psychologism is not widely held amongst logicians today, but it does have some high-profile defenders, for example Dov Gabbay.
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For Frege asserting the negation of a claim serves roughly the same role as denying a claim does in Aristotle. Other Western logicians such as Kant and Hegel answer, ultimately three; you can affirm, deny or make merely limiting affirmations, which transcend both affirmation and denial. In Indian logic, four logical qualities have been the norm, and Nagarjuna is sometimes interpreted as arguing for five.
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Materna was introduced to philosophy and logic by his father, Miloš Materna (9 April 1892 – 4 August 1951), a member of a group of interwar and postwar Czech logicians propagating and popularizing neopositivism in Czechoslovakia.Materna, Pavel (1998) "Miloš Materna" IN Slovník českých filozofů [Dictionary of Czech Philosophers]. Brno: Masaryk University Press, p. 383. . In 1949, he enrolled at the Faculty of Arts of Charles University to study philosophy and psychology.
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Goodman reports the puzzle came back to him from various directions, including a 1936 Warsaw Logicians' meeting via Carnap; some echo versions were corrupted by joining B two utterances into a single one, which make the puzzle unsolvable. Some years later, Goodman heard about the #Fork in the road variant; having scruples about counterfactuals, he devised a non- subjunctive, non-contrary-to-fact question that can be asked.
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In their monograph, Dijkstra and Scholten use the three inference rules Leibniz, Substitution, and Transitivity. However, Dijkstra/Scholten system is not a logic, as logicians use the word. Some of their manipulations are based on the meanings of the terms involved, and not on clearly presented syntactical rules of manipulation. The first attempt at making a real logic out of it appeared in A Logical Approach to Discrete Math.
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The logicians (School of Names) were concerned with logic, paradoxes, names and actuality (similar to Confucian rectification of names). The logician Hui Shi was a friendly rival to Zhuangzi, arguing against Taoism in a light-hearted and humorous manner. Another logician, Gongsun Long, originated the famous When a White Horse is Not a Horse dialogue. This school did not thrive because the Chinese regarded sophistry and dialectic as impractical.
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In TeX, the turnstile symbols \vDash and \models are obtained from the commands `\vDash` and `\models` respectively. In Unicode it is encoded at In LaTeX there is the turnstile package, which issues this sign in many ways, including the double turnstile, and is capable of putting labels below or above it, in the correct places. The article A Tool for Logicians is a tutorial on using this package.
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Schröder's early work on formal algebra and logic was written in ignorance of the British logicians George Boole and Augustus De Morgan. Instead, his sources were texts by Ohm, Hankel, Hermann Grassmann, and Robert Grassmann (Peckhaus 1997: 233-296). In 1873, Schröder learned of Boole's and De Morgan's work on logic. To their work he subsequently added several important concepts due to Charles Sanders Peirce, including subsumption and quantification.
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He believed that ethical norms had been invented to rectify mankind. Other philosophers and logicians such as Guanzi, Mozi, and Gongsun Long developed their own theories regarding the rectification. Li in itself can be seen as the root of all this propriety and social etiquette discussed in the rectification of names as the cure to society's problems and the solution to a moral and efficient government and society.
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A scientific model is a simplified abstract view of a complex reality. A scientific model represents empirical objects, phenomena, and physical processes in a logical way. Attempts to formalize the principles of the empirical sciences use an interpretation to model reality, in the same way logicians axiomatize the principles of logic. The aim of these attempts is to construct a formal system for which reality is the only interpretation.
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Muslim logicians had inherited Greek ideas after they had invaded and conquered Egypt and the Levant. Their translations and commentaries on these ideas worked their way through the Arab West into Iberia and Sicily, which became important centers for this transmission of ideas. From the 11th to the 13th century, many schools dedicated to the translation of philosophical and scientific works from Classical Arabic to Medieval Latin were established in Iberia.
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In the Hebrew versions, the Treatise is called The words of Logic which describes the bulk of the work. The author explains the technical meaning of the words used by logicians. The Treatise duly inventories the terms used by the logician and indicates what they refer to. The work proceeds rationally through a lexicon of philosophical terms to a summary of higher philosophical topics, in 14 chapters corresponding to Maimonides's birthdate of 14 Nissan.
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7: pp.37-62. The Sanskrit name has been reconstructed as either Prajñāpradīpa or Janāndeepa (where Janāndeepa may or may not be a Prakrit corruption or a poor inverse- translation, for example). According to Ames (1993: p. 210), Bhāviveka was one of the first Buddhist logicians to employ the "formal syllogism" (, ) of Indian logic in expounding the Mādhyamaka which he employed to considerable effect in his commentary to Nāgārjuna's Mūlamadhyamakakārikā, entitled the Prajñāpradīpa.
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Prior to Linnaean taxonomy, animals were classified according to their mode of movement. Linnaeus's use of binomial nomenclature was anticipated by the theory of definition used in Scholasticism. Scholastic logicians and philosophers of nature defined the species man, for example, as Animal rationalis, where animal was considered a genus and rationalis (Latin for "rational") the characteristic distinguishing man from all other animals. Treating animal as the immediate genus of the species man, horse, etc.
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Arabic logicians had inherited Greek ideas after they had invaded and conquered Egypt and the Levant. Their translations and commentaries on these ideas worked their way through the Arab West into Spain and Sicily, which became important centers for this transmission of ideas. Western Arabic translations of Greek works (found in Iberia and Sicily) originates in the Greek sources preserved by the Byzantines. These transmissions to the Arab West took place in two main stages.
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The Posterior Analytics (Latin: Analytica Posteriora) deals with demonstration, definition, and scientific knowledge. :5. The Topics (Latin: Topica) treats issues in constructing valid arguments, and inference that is probable, rather than certain. It is in this treatise that Aristotle mentions the Predicables, later discussed by Porphyry and the scholastic logicians. :6. The Sophistical Refutations (Latin: De Sophisticis Elenchis) gives a treatment of logical fallacies, and provides a key link to Aristotle's work on rhetoric.
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When scientists attempt to formalize the principles of the empirical sciences, they use an interpretation to model reality, in the same way logicians axiomatize the principles of logic. The aim of these attempts is to construct a formal system that will serve as a conceptual model of reality. Predictions or other statements drawn from such a formal system mirror or map the real world only insofar as these scientific models are true.
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On the other hand, al-Ghazali (1058–1111; and, in modern times, Abu Muhammad Asem al-Maqdisi) argued that Qiyas refers to analogical reasoning in a real sense and categorical syllogism in a metaphorical sense. Other Islamic scholars at the time, however, argued that the term Qiyas refers to both analogical reasoning and categorical syllogism in a real sense.Wael B. Hallaq (1993), Ibn Taymiyya Against the Greek Logicians, p. 48. Oxford University Press, .
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""se dit aussi au figuré, pour irrégulier, bizarre, inégale." Le Dictionnaire de l'Académie française (1762) Jean-Jacques Rousseau, who was a musician and composer as well as philosopher, wrote in 1768 in the Encyclopédie: "Baroque music is that in which the harmony is confused, and loaded with modulations and dissonances. The singing is harsh and unnatural, the intonation difficult, and the movement limited. It appears that term comes from the word 'baroco' used by logicians.
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Jaśkowski is considered to be one of the founders of natural deduction, which he discovered independently of Gerhard Gentzen in the 1930s. Gentzen's approach initially became more popular with logicians because it could be used to prove the cut-elimination theorem. However, Jaśkowski's is closer to the way that proofs are done in practice. He was also one of the first to propose a formal calculus of inconsistency- tolerant (or paraconsistent) logic.
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He built on and reinterpreted the work of Dignaga, the pioneer of Buddhist Logic, and was very influential among Brahman logicians as well as Buddhists. His theories became normative in Tibet and are studied to this day as a part of the basic monastic curriculum. Other Buddhist monks that visited Indonesia were Atisha, Dharmapala, a professor of Nalanda, and the South Indian Buddhist Vajrabodhi. Srivijaya was the largest Buddhist empire ever formed in Indonesian history.
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Albert also authored commentaries on Ars Vetus, a set of twenty-five Quaestiones logicales (c. 1356) that involved semantical problems and the status of logic, and Quaestiones on the Posterior Analytics. Albert explored in a series of disputed questions the status of logic and semantics, as well as the theory of reference and truth. Albert was influenced by English logicians and was influential in the diffusion of terminist logic in central Europe.
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Theodore Hailperin showed much earlier that Boole had used the correct mathematical definition of independence in his worked out problems. Boole's work and that of later logicians initially appeared to have no engineering uses. Claude Shannon attended a philosophy class at the University of Michigan which introduced him to Boole's studies. Shannon recognised that Boole's work could form the basis of mechanisms and processes in the real world and that it was therefore highly relevant.
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Many early Islamic philosophers and logicians discussed the liar paradox. Their work on the subject began in the 10th century and continued to Athīr al- Dīn al-Abharī and Nasir al-Din al-Tusi of the middle 13th century and beyond. Although the Liar paradox has been well known in Greek and Latin traditions, the works of Arabic scholars have only recently been translated into English. Each group of early Islamic philosophers discussed different problems presented by the paradox.
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Harvey Friedman conjectured, "Every theorem published in the Annals of Mathematics whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in elementary arithmetic.". The form of elementary arithmetic referred to in this conjecture can be formalized by a small set of axioms concerning integer arithmetic and mathematical induction. For instance, according to this conjecture, Fermat's Last Theorem should have an elementary proof; Wiles' proof of Fermat's Last Theorem is not elementary.
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Hume offers no solution to the problem of induction himself. He prompts other thinkers and logicians to argue for the validity of induction as an ongoing dilemma for philosophy. A key issue with establishing the validity of induction is that one is tempted to use an inductive inference as a form of justification itself. This is because people commonly justify the validity of induction by pointing to the many instances in the past when induction proved to be accurate.
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The > mind-body correlations as formulated at present, do not admit of spatial > correlation, so they reduce to matters of simple correlation in time. The > need for identification is no less urgent in this case (p. 16, quoted in > Place [unpublished]). The barrier to the acceptance of any such vision of the mind, according to Place, was that philosophers and logicians had not yet taken a substantial interest in questions of identity and referential identification in general.
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The predicables (Lat. praedicabilis, that which may be stated or affirmed, sometimes called quinque voces or five words) is, in scholastic logic, a term applied to a classification of the possible relations in which a predicate may stand to its subject. The list given by the schoolmen and generally adopted by modern logicians is based on the original fourfold classification given by Aristotle (Topics, a iv. 101 b 17–25): definition (horos), genus (genos), property (idion), accident (sumbebekos).
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Cognitive psychology involves the study of cognition, including mental processes underlying perception, learning, problem solving, reasoning, thinking, memory, attention, language, and emotion. Classical cognitive psychology has developed an information processing model of mental function, and has been informed by functionalism and experimental psychology. Cognitive science is an interdisciplinary research enterprise that involves cognitive psychologists, cognitive neuroscientists, artificial intelligence, linguists, human–computer interaction, computational neuroscience, logicians and social scientists. Computational models are sometimes used to simulate phenomena of interest.
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Tarski, the most prominent member of the Lwów–Warsaw School, has been ranked as one of the four greatest logicians of all time, along with Aristotle, Gottlob Frege, and Kurt Gödel.Feferman & Feferman, p. 1 The school's work was interrupted by the outbreak of World War II. Despite this, its members went on to fundamentally influence modern science, notably mathematics and logic, in the post-war period. Tarski's description of semantic truth, for instance, has revolutionized logic and philosophy.
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The theological term "invincible ignorance" should not be confused with the logical term Invincible ignorance fallacy. When and how the term was taken by logicians to refer to the very different state of persons who willfully refuse to attend to evidence remains unclear, but one of its first uses was in the 1959 book Fallacy: The Counterfeit of Argument by W. Ward Fearnside and William B. Holther.Fearnside, W. Ward and William B. Holther, Fallacy: The Counterfeit of Argument, 1959. .
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The 1930 Königsberg conference was a joint meeting of three academic societies, with many of the key logicians of the time in attendance. Carnap, Heyting, and von Neumann delivered one-hour addresses on the mathematical philosophies of logicism, intuitionism, and formalism, respectively (Dawson 1996, p. 69). The conference also included Hilbert's retirement address, as he was leaving his position at the University of Göttingen. Hilbert used the speech to argue his belief that all mathematical problems can be solved.
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The real-world use and practical application of fa were vital. Yet fa as models were also used in later Mohist logic as principles used in deductive reasoning. As classical Chinese philosophical logic was based on analogy rather than syllogism, fa were used as benchmarks to determine the validity of logical claims through comparison. There were three fa in particular that were used by these later Mohists (or "Logicians") to assess such claims, which were mentioned earlier.
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Codd's Theorem is notable since it establishes the equivalence of two syntactically quite dissimilar languages: relational algebra is a variable-free language, while relational calculus is a logical language with variables and quantification. Relational calculus is essentially equivalent to first-order logic, and indeed, Codd's Theorem had been known to logicians since the late 1940s. Query languages that are equivalent in expressive power to relational algebra were called relationally complete by Codd. By Codd's Theorem, this includes relational calculus.
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For example, in Finnish, join vettä, "I drank (some) water", the word vesi, "water", is in the partitive case. The related sentence join veden, "I drank (the) water", using the accusative case instead, assumes that there was a specific countable portion of water that was completely drunk. The work of logicians like Godehard Link and Manfred Krifka established that the mass/count distinction can be given a precise, mathematical definition in terms of quantization and cumulativity.
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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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Bhartrhari thus rejected the difference posited between the ontological and the linguistic by logicians. His concept of shabda-brahman which identified linguistic performance and creation itself ran parallel to the Greek concept of logos. Language philosophy in Medieval India was dominated by the dispute of the "naturalists" to the Mimamsa school, notably defended by Kumarila, who held that shabda designates the actual phonetic utterance, and the Sphota school, defended by Mandana Mishra, which identifies sphota and shabda as a mystical "indivisible word-whole".
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The Ajivikas had a fully elaborate philosophy, produced by its scholars and logicians, but those texts are lost. Their literature evolved over the centuries, like other traditions of Indian philosophy, through the medieval era. The Pali and Prakrit texts of Buddhism and Jainism suggest that Ajivika theories were codified, some of which were quoted in commentaries produced by Buddhist and Jaina scholars. The main texts of the Ajivikas included the ten Purvas (eight Mahanimittas, two Maggas) and the Onpatu Katir.
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Suhrawardi's Illuminationist project was to have far- reaching consequences for Islamic philosophy in Shi'ite Iran. His teachings had a strong influence on subsequent esoteric Iranian thought and the idea of “Decisive Necessity” is believed to be one of the most important innovations in the history of logical philosophical speculation, stressed by the majority of Muslim logicians and philosophers. In the 17th century, it was to initiate an Illuminationist Zoroastrian revival in the figure of the 16th century sage Azar Kayvan.
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76 a.2 and discussion of the concept can be found as far back as Origen (3rd century). When and how the term was taken by logicians to refer to the very different state of persons who deliberately refuse to attend to evidence remains unclear, but one of its first uses was in the book Fallacy: The Counterfeit of Argument by W. Ward Fearnside and William B. HoltherFearnside, W. Ward and William B. Holther, Fallacy: The Counterfeit of Argument, 1959. . in 1959.
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Boethius (c. 475 – 526) contributed an effort to make the ancient Aristotelian logic more accessible. While his Latin translation of Prior Analytics went primarily unused before the 12th century, his textbooks on the categorical syllogism were central to expanding the syllogistic discussion. Rather than in any additions that he personally made to the field, Boethius' logical legacy lies in his effective transmission of prior theories to later logicians, as well as his clear and primarily accurate presentations of Aristotle's contributions.
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According to the Routledge Encyclopedia of Philosophy: Important developments made by Muslim logicians included the development of "Avicennian logic" as a replacement of Aristotelian logic. Avicenna's system of logic was responsible for the introduction of hypothetical syllogism, temporal modal logic and inductive logic. Other important developments in early Islamic philosophy include the development of a strict science of citation, the isnad or "backing", and the development of a method to disprove claims, the ijtihad, which was generally applied to many types of questions.
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As an algebraist, Blok "was recognised by the modal logic community as one of the most influential modal logicians" by the end of the 1970s. He published many papers in the Reports on Mathematical Logic, served as a member on their editorial board, and was one of their guest editors. Along with Don Pigozzi, Wim Blok co-authored the monograph Algebraizable Logics which began the field now known as abstract algebraic logic. He died in a car accident on November 30, 2003.
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Bourbaki's members were mathematicians as opposed to logicians, and therefore the collective had a limited interest in mathematical logic. As Bourbaki's members themselves said of the book on set theory, it was written "with pain and without pleasure, but we had to do it." Dieudonné personally remarked elsewhere that ninety-five percent of mathematicians "don't care a fig" for mathematical logic. In response, logician Adrian Mathias harshly criticized Bourbaki's foundational framework, noting that it did not take Gödel's results into account.
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According to Georges Dreyfus, while Western logic tends to be focused on formal validity and deduction: > The concern of Indian "logicians" is quite different. They intend to provide > a critical and systematic analysis of the diverse means of correct cognition > that we use practically in our quest for knowledge. In this task, they > discuss the nature and types of pramana. Although Indian philosophers > disagree on the types of cognition that can be considered valid, most > recognize perception and inference as valid.
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For Eriugena, philosophy or reason is first, primitive; authority or religion is secondary, derived. Eriugena's influence was greater with mystics than with logicians, but he was responsible for a revival of philosophical thought which had remained largely dormant in western Europe after the death of Boethius. Leszek Kołakowski, a Polish Marx scholar, has mentioned Eriugena as one of the primary influences on Hegel's, and therefore Marx's, dialectical form. In particular, he called De Divisione Naturae a prototype of Hegel's Phenomenology of Spirit..
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Within the system of propositional logic, no proposition or variable carries any semantic content. The moment any proposition or variable takes on semantic content, the problem arises again because semantic content runs outside the system. Thus, if the solution is to be said to work, then it is to be said to work solely within the given formal system, and not otherwise. Some logicians (Kenneth Ross, Charles Wright) draw a firm distinction between the conditional connective and the implication relation.
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L. T. F. Gamut was a collective pseudonym for the Dutch logicians Johan van Benthem, Jeroen Groenendijk, Dick de Jongh, Martin Stokhof and Henk Verkuyl.Preface to Logic, Language and Meaning, by L. T. F. Gamut, University of Chicago Press, 1991. Gamut stands for the Dutch universities of Groningen (G), Amsterdam (am), and Utrecht (ut), then the affiliations of the authors. The initials L. T. F. stand for the discussed topics, respectively, Logic (Dutch: Logica), Language (Dutch: Taal) and Philosophy (Dutch: Filosofie).
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The formally sophisticated treatment of modern logic descends from the Greek tradition, being informed from the transmission of Aristotelian logic, which was then further developed by Islamic logicians. The Indian tradition also continued into the early modern period. The native Chinese tradition did not survive beyond antiquity, though Indian logic was later adopted in medieval China. As a number of other disciplines of formal science rely heavily on mathematics, they did not exist until mathematics had developed into a relatively advanced level.
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An Essay Concerning Human Understanding IV.5, 1-8. This view, known as psychologism, was taken to the extreme in the nineteenth century, and is generally held by modern logicians to signify a low point in the decline of logic before the twentieth century. Modern semantics is in some ways closer to the medieval view, in rejecting such psychological truth-conditions. However, the introduction of quantification, needed to solve the problem of multiple generality, rendered impossible the kind of subject- predicate analysis that underlies medieval semantics.
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The Survey sent him to Europe five times, first in 1871 as part of a group sent to observe a solar eclipse. There, he sought out Augustus De Morgan, William Stanley Jevons, and William Kingdon Clifford, British mathematicians and logicians whose turn of mind resembled his own. From 1869 to 1872, he was employed as an assistant in Harvard's astronomical observatory, doing important work on determining the brightness of stars and the shape of the Milky Way.Moore, Edward C., and Robin, Richard S., eds.
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Over the course of the 20th century, a number of Polish logicians and mathematicians contributed to this "Polish mereology." Even though Polish mereology is now only of historical interest, the word "mereology" endures as the name of a collection of first order theories relating parts to their respective wholes. These theories, unlike set theory, can be proved sound and complete. Nearly all work that has appeared since 1970 under the heading of mereology descends from the 1940 calculus of individuals of Henry Leonard and Nelson Goodman.
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Hilbert was initially a deductivist, but, as may be clear from above, he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic. Later, he held the opinion that there was no other meaningful mathematics whatsoever, regardless of interpretation. Other formalists, such as Rudolf Carnap, Alfred Tarski, and Haskell Curry, considered mathematics to be the investigation of formal axiom systems. Mathematical logicians study formal systems but are just as often realists as they are formalists.
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Cognitive psychology studies cognition, the mental processes underlying mental activity. Perception, learning, problem solving, reasoning, thinking, memory, attention, language and emotion are areas of research. Classical cognitive psychology is associated with a school of thought known as cognitivism, whose adherents argue for an information processing model of mental function, informed by functionalism and experimental psychology. On a broader level, cognitive science is an interdisciplinary enterprise of cognitive psychologists, cognitive neuroscientists, researchers in artificial intelligence, linguists, human–computer interaction, computational neuroscience, logicians and social scientists.
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But when we use the words "if" and "then" we generally mean to assert that there is some relation between the antecedent and the consequent. What is the nature of that relationship? Relevance (or relevant) logicians take the view that, in addition to saying that the consequent cannot be false while the antecedent is true, the antecedent must be "relevant" to the consequent. At least initially, this means that there must be at least some terms (or variables) that appear in both the antecedent and the consequent.
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Another of medieval logic's first contributors from the Latin West, Peter Abelard (1079–1142), gave his own thorough evaluation of the syllogism concept and accompanying theory in the Dialectica—a discussion of logic based on Boethius' commentaries and monographs. His perspective on syllogisms can be found in other works as well, such as Logica Ingredientibus. With the help of Abelard's distinction between de dicto modal sentences and de re modal sentences, medieval logicians began to shape a more coherent concept of Aristotle's modal syllogism model.
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The Lwów–Warsaw School () was a Polish school of thought founded by Kazimierz Twardowski in 1895 in Lemberg (Polish name: Lwów), Austro-Hungary (now Lviv, Ukraine). Though its members represented a variety of disciplines, from mathematics through logic to psychology, the Lwów–Warsaw School is widely considered to have been a philosophical movement.Jan Woleński, Filozoficzna szkoła lwowsko-warszawska, Warsaw, PWN, 1985. It has produced some of the leading logicians of the twentieth century such as Jan Lukasiewicz, Stanislaw Lesniewski, and Alfred Tarski, among others.
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Bertrand Russell (1872–1970) led the British "revolt against idealism" in the early 1900s, along with G. E. Moore. He was influenced by Gottlob Frege, and was the mentor of Ludwig Wittgenstein. He is widely held to be one of the 20th century's premier logicians."Bertrand Russell" at the Stanford Encyclopedia of Philososophy Retrieved May 15, 2010 He co-authored, with Alfred North Whitehead, Principia Mathematica, an attempt to derive all mathematical truths from a set of axioms using rules of inference in symbolic logic.
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He has also been active in the fields of philosophy of science, logical structures in natural language (generalized quantifiers, categorial grammar, substructural proof theory), dynamic logic and update logic, and applications of logic to game theory, as well as applications of game theory to logic (game semantics). Van Benthem is a member of the group collectively publishing under the pseudonym L. T. F. Gamut. He has also taught in China. He made an effort to encourage and organize international collaboration between Chinese and Western logicians.
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Although it is essentially the proof of a single theorem, aimed at specialists in the area, the book is written in a readable style that introduces the reader to many important topics in finite model theory and the theory of random graphs. Reviewer Valentin Kolchin, himself the author of another book on random graphs, writes that the book is "self-contained, easily read, and is distinguished by elegant writing", recommending it to probability theorists and logicians. Reviewer Alessandro Berarducci calls the book "beautifully written" and its subject "fascinating".
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The following very helpful passage by philosopher James Franklin gives some hint as to the history of the Porphyrian tree: :In medieval education, the standard introduction to Aristotle's works was via Porphyry's Isagoge, and division entered the educated consciousness in the form of 'Porphyry's Tree'. It is not clear that Porphyry himself, in the relevant passage,Franklin's note: "Porphyry, Isagoge, trans. E. W. Warren (Toronto: Pontifical Institute of Medieval Studies, 1975), 34." went any further than Aristotle in recommending division. But his brief comment was developed into the Tree by medieval logicians.
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These results confirm the validity of the argument A Some arguments need first-order predicate logic to reveal their forms and they cannot be tested properly by truth tables forms. Consider the argument A1: > Some mortals are not Greeks > Some Greeks are not men > Not every man is a logician > Therefore Some mortals are not logicians > To test this argument for validity, construct the corresponding conditional C1 (you will need first-order predicate logic), negate it, and see if you can derive a contradiction from it. If you succeed, then the argument is valid.
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Model theory has a different scope that encompasses more arbitrary theories, including foundational structures such as models of set theory. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic. For a given theory in model theory, a structure is called a model if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a semantic model when one discusses the notion in the more general setting of mathematical models. Logicians sometimes refer to structures as interpretations.
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In Islam, a criticism is raised, wherein it is argued that "from the juristic standpoint, obliterating the distinctions between God and the universe necessarily entails that in effect there can be no Sharia, since the deontic nature of the Law presupposes the existence of someone who commands (amir) and others who are the recipients of the command (ma'mur), namely God and his subjects."Aḥmad ibn ʻAbd al-Ḥalīm Ibn Taymīyah, Wael B. Hallaq, Ibn Taymiyya against the Greek logicians, 1993, xxvi. In 1996, Pastor Bob BurridgeGenevan Institute for Reformed Studies.Homepage of Bob Burridge .
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Tarski's student, Vaught, has ranked Tarski as one of the four greatest logicians of all time -- along with Aristotle, Gottlob Frege, and Kurt Gödel. However, Tarski often expressed great admiration for Charles Sanders Peirce, particularly for his pioneering work in the logic of relations. Tarski produced axioms for logical consequence and worked on deductive systems, the algebra of logic, and the theory of definability. His semantic methods, which culminated in the model theory he and a number of his Berkeley students developed in the 1950s and 60s, radically transformed Hilbert's proof-theoretic metamathematics.
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From 1970 to 1973 Smith studied Mathematics and Philosophy at the University of Oxford. He obtained his PhD from the University of Manchester in 1976 for a dissertation on ontology and reference in Husserl and Frege. The dissertation was supervised by Wolfe Mays. Among the cohort of graduate students supervised by Mays in Manchester were Kevin Mulligan (Geneva/Lugano), and Peter Simons (Trinity College, Dublin), who shared with Smith an interest in analytic metaphysics and in the contributions of certain turn-of-the-century Continental philosophers and logicians to central issues of analytic philosophy.
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Schröder's influence on the early development of the predicate calculus, mainly by popularising C. S. Peirce's work on quantification, is at least as great as that of Frege or Peano. For an example of the influence of Schröder's work on English-speaking logicians of the early 20th century, see Clarence Irving Lewis (1918). The relational concepts that pervade Principia Mathematica are very much owed to the Vorlesungen, cited in Principia's Preface and in Bertrand Russell's Principles of Mathematics. Frege (1960) dismissed Schröder's work, and admiration for Frege's pioneering role has dominated subsequent historical discussion.
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The logicians Aaleyah and Isko are sitting in their dark office wondering whether or not it is raining outside. Now, none of them actually knows, but Aaleyah knows something about her friend Yu Yan, namely that Yu Yan wears her red coat only if it is raining. Isko does not know this, but he just saw Yu Yan, and noticed that she was wearing her red coat. Even though none of them knows whether or not it is raining, it is distributed knowledge amongst them that it is raining.
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Ruth Barcan Marcus (; born Ruth Charlotte Barcan; 2 August 1921 - 19 February 2012) was an American academic philosopher and logician best known for her work in modal and philosophical logic. She developed the first formal systems of quantified modal logic and in so doing introduced the schema or principle known as the Barcan formula. (She would also introduce the now standard "box" operator for necessity in the process). Marcus, who originally published as Ruth C. Barcan, was, as Don Garrett notes "one of the twentieth century’s most important and influential philosopher-logicians".
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A set of axioms should be consistent; it should be impossible to derive a contradiction from the axiom. A set of axioms should also be non- redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom. It was the early hope of modern logicians that various branches of mathematics, perhaps all of mathematics, could be derived from a consistent collection of basic axioms. An early success of the formalist program was Hilbert's formalization of Euclidean geometry,For more, see Hilbert's axioms.
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An early formal system of temporal logic was studied by Avicenna.History of logic: Arabic logic, Encyclopædia Britannica. Although he did not develop a real theory of temporal propositions, he did study the relationship between temporalis and the implication. Avicenna's work was further developed by Najm al-Dīn al-Qazwīnī al-Kātibī and became the dominant system of Islamic logic until modern times. Avicennian logic also influenced several early European logicians such as Albertus MagnusRichard F. Washell (1973), "Logic, Language, and Albert the Great", Journal of the History of Ideas 34 (3), pp.
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The Löwenheim–Skolem theorem dealt a first blow to this hope, as it implies that a first-order theory which has an infinite model cannot be categorical. Later, in 1931, the hope was shattered completely by Gödel's incompleteness theorem. Many consequences of the Löwenheim–Skolem theorem seemed counterintuitive to logicians in the early 20th century, as the distinction between first-order and non-first-order properties was not yet understood. One such consequence is the existence of uncountable models of true arithmetic, which satisfy every first-order induction axiom but have non-inductive subsets.
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This caused Russell to analyse classes, for it was known that given any number of elements, the number of classes they result in is greater than their number. This in turn led to the discovery of a very interesting class, namely, the class of all classes. It contains two kinds of classes: those classes that contain themselves, and those that do not. Consideration of this class led him to find a fatal flaw in the so-called principle of comprehension, which had been taken for granted by logicians of the time.
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The first volume of Sigwart's principal work, Logik, was published in 1873 and took an important place among contributions to logical theory in the late nineteenth century. In the preface to the first edition, Sigwart explains that he makes no attempt to appreciate the logical theories of his predecessors; he intended to construct a theory of logic, complete in itself. The Logik represents the results of a long and careful study not only of German but also of English logicians. In 1895 an English translation by Helen Dendy was published in London.
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The philosophy of the Logicians is often considered to be akin to those of the sophists or of the dialecticians. Joseph Needham notes that their works have been lost, except for the partially preserved Gongsun Longzi, and the paradoxes of Chapter 33 of the Zhuangzi. Needham considers the disappearance of the greater part of Gongsun Longzi one of the worst losses in the ancient Chinese books, as what remains is said to reach the highest point of ancient Chinese philosophical writing. Birth places of notable Chinese philosophers from Hundred Schools of Thought in Zhou Dynasty.
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According to Watts, and in keeping with logicians of his day, Watts defined logic as an art (see liberal arts), as opposed to a science. Throughout Logic, Watts revealed his high conception of logic by stressing the practical side of logic, rather than the speculative side. According to Watts, as a practical art, logic can be really useful in any inquiry, whether it is an inquiry in the arts, or inquiry in the sciences, or inquiry of an ethical kind. Watts' emphasis on logic as a practical art distinguishes his book from others.
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Possible worlds are one of the foundational concepts in modal and intensional logics. Formulas in these logics are used to represent statements about what might be true, what should be true, what one believes to be true and so forth. To give these statements a formal interpretation, logicians use structures containing possible worlds. For instance, in the relational semantics for classical propositional modal logic, the formula \Diamond P (read as "possibly P") is actually true iff P is true at some world which is accessible from the actual world.
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According to the Routledge Encyclopedia of Philosophy: Important developments made by Muslim logicians included the development of "Avicennian logic" as a replacement of Aristotelian logic. Avicenna's system of logic was responsible for the introduction of hypothetical syllogism, temporal modal logic and inductive logic. Other important developments in early Islamic philosophy include the development of a strict science of citation, the isnad or "backing", and the development of a scientific method of open inquiry to disprove claims, the ijtihad, which could be generally applied to many types of questions.
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A version of the paradox occurs already in chapter 9 of Thomas Bradwardine’s Insolubilia.Bradwardine, T. (2010), Insolubilia, Latin text and English translation by Stephen Read, Peeters, Leuven. In the wake of the modern discussion of the paradoxes of self-reference, the paradox has been rediscovered (and dubbed with its current name) by the US logicians and philosophers David Kaplan and Richard Montague,Kaplan, D. and Montague, R. (1960), 'A Paradox Regained', Notre Dame Journal of Formal Logic 1, pp. 79–90. and is now considered an important paradox in the area.
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Her trustees decided to sell the Paris property and acquire an ample plot on the Montagne Sainte-Geneviève (rue de la Montagne- Sainte-Geneviève / rue Descartes), right in the Latin Quarter, and build the college anew. The first stone, laid 12 April 1309, was for the college chapel. Provision was made also for the scholars' support, 4 Paris sous weekly for the artists, 6 for the logicians and 8 for the theologians. These allowances were to continue until the graduates held benefices of the value respectively of 30, 40 and 60 livres.
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He studied first as an unenrolled student at Jagiellonian University in Kraków, attending mostly lectures on mathematics and physics; then architecture in Lviv and Darmstadt, to finally settle for studies in philosophy and classical philology at the University of Lviv. His professors were some of the most esteemed philosophers, logicians and mathematicians of his time: Kazimierz Twardowski, Jan Łukasiewicz, Władysław Witwicki and philologist Stanisław Witkowski. He received his PhD with the thesis Utilitarianism in the Ethics of Mill and Spencer in 1912. After graduation, he taught classical languages at Warsaw's Mikołaj Rey Gymnasium (secondary school).
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In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.
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He received his B.A. in mathematics from Oberlin College in 1930, and his Ph.D. in philosophy from Harvard University in 1932. His thesis supervisor was Alfred North Whitehead. He was then appointed a Harvard Junior Fellow, which excused him from having to teach for four years. During the academic year 1932–33, he travelled in Europe thanks to a Sheldon fellowship, meeting Polish logicians (including Stanislaw Lesniewski and Alfred Tarski) and members of the Vienna Circle (including Rudolf Carnap), as well as the logical positivist A. J. Ayer.
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As the Cybermen emerge, Klieg reveals that he and Kaftan belong to the Brotherhood of Logicians, a cult that possess great intelligence but no physical power. He believes the Cybermen will be grateful for their revival and will ally themselves with the Brotherhood to conquer the universe. After they awaken, the Cybermen revive their leader, the Cyber-Controller, and take the group as prisoners. The Doctor realizes that the tombs are actually an elaborate trap, with the Cybermen keeping themselves frozen until they were revived and rebuild their invasion force to conquer Earth.
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The fruit of that inspiration is non-Frege logic, one of the key achievements of the post-war Polish logic. One student of Suszko and a colleague of Wolniewicz was a logician Mieczysław Omyła, who continued both logical and ontological work of Suszko and referenced Wolniewicz's situational ontology more than once. Situational ontology also inspires mathematical papers on conditionally distributive lattices by Jan Zygmunt and Jacek Hawranek. Wolniewicz's papers were also met with interest from philosophers and logicians abroad, as evidenced by multiple reviews of his English papers in "Mathematical Review".
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This difficulty, which seems to be inherent in the very nature of truth and falsehood, is one with which I do not know how to deal with satisfactorily. ...I therefore leave this question to the logicians with the above brief indication of a difficulty. (§ 52) Consider e.g. "A differs from B". The constituents of this proposition are simply A, difference and B. The proposition relates A and B, using the words "is ... from" in "A is different from B". But if we represent this contribution by words for relations, as e.g.
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Sextus Empiricus argued that "person" could not be precisely defined. He debunks various definitions of “human” given by philosophical schools, by showing that they are speculative and disagree with each other, that they identify properties (many not even definitive anyway) rather than the property-holder, and that none of these definitions seem to include every human and exclude every non-human.Against the Logicians I: 263-282; Outlines of Pyrrhonism II: 22-28 This debunking is similar to the Buddhist arguments against the existence of the “person.” The person is said to lack identifiable entity-hood.
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Although his work had considerable impact in the area of Artificial intelligence and law, Pollock was not himself interested much in jurisprudence or theories of legal reasoning, and he never acknowledged the inheritance of defeasible reasoning through H.L.A. Hart. Pollock also held informal logicians and scholars of rhetoric at a distance, though defeasible reasoning has natural affinities in argument (logic). Pollock's "undercutting defeat" and "rebutting defeat" are now fixtures in the defeasible reasoning literature. He later added "self-defeat" and other kinds of defeat mechanisms, but the original distinction remains the most popular.
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In mathematical logic, the Lindenbaum–Tarski algebra (or Lindenbaum algebra) of a logical theory T consists of the equivalence classes of sentences of the theory (i.e., the quotient, under the equivalence relation ~ defined such that p ~ q exactly when p and q are provably equivalent in T). That is, two sentences are equivalent if the theory T proves that each implies the other. The Lindenbaum–Tarski algebra is thus the quotient algebra obtained by factoring the algebra of formulas by this congruence relation. The algebra is named for logicians Adolf Lindenbaum and Alfred Tarski.
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New York: McGraw Hill. Pp. 5 - 28 In his article "Charity Begins at Home" in Informal Logic Johnson combines and creates unified form of the 'Principle of Charity' which he found to exist in four other forms in the following works: Thomas's Practical Reasoning in Natural Language (1973), Baum's Logic (1975) and in Scriven's Reasoning (1976). In doing so Johnson created a more developed 'Principle of Charity' which Informal Logicians could reference.1981 "Charity Begins at Home: Some Reflections on the Principle of Charity," Informal Logic Newsletter, Vol.
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Until the coming of the 20th century, later logicians followed Aristotelian logic, which includes or assumes the law of the excluded middle. The 20th century brought back the idea of multi-valued logic. The Polish logician and philosopher Jan Łukasiewicz began to create systems of many-valued logic in 1920, using a third value, "possible", to deal with Aristotle's paradox of the sea battle. Meanwhile, the American mathematician, Emil L. Post (1921), also introduced the formulation of additional truth degrees with n ≥ 2, where n are the truth values.
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Idealist notions have been propounded by the Vedanta schools of thought, which use the Vedas, especially the Upanishads as their key texts. Idealism was opposed by dualists Samkhya, the atomists Vaisheshika, the logicians Nyaya, the linguists Mimamsa and the materialists Cārvāka. There are various sub schools of Vedanta, like Advaita Vedanta (non-dual), Vishishtadvaita and Bhedabheda Vedanta (difference and non-difference). The schools of Vedanta all attempt to explain the nature and relationship of Brahman (universal soul or Self) and Atman (individual self), which they see as the central topic of the Vedas.
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An example traditionally used by logicians contrasting sufficient and necessary conditions is the statement "If there is fire, then oxygen is present". An oxygenated environment is necessary for fire or combustion, but simply because there is an oxygenated environment does not necessarily mean that fire or combustion is occurring. While one can infer that fire stipulates the presence of oxygen, from the presence of oxygen the converse "If there is oxygen present, then fire is present" cannot be inferred. All that can be inferred from the original proposition is that "If oxygen is not present, then there cannot be fire".
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Programmers, statisticians or logicians are concerned in their work with the main operational or technical significance of a concept which is specifiable in objective, quantifiable terms. They are not primarily concerned with all kinds of imaginative frameworks associated with the concept, or with those aspects of the concept which seem to have no particular functional purpose – however entertaining they might be. However, some of the qualitative characteristics of the concept may not be quantifiable or measurable at all, at least not directly. The temptation exists to ignore them, or try to infer them from data results.
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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. However, they were nonproductive and not integrated into Chinese science or mathematics. The Mohist school of Chinese philosophy contained an approach to logic and argumentation that stresses rhetorical analogies over mathematical reasoning, and is based on the three fa, or methods of drawing distinctions between kinds of things. One of the schools that grew out of Mohism, the Logicians, are credited by some scholars for their early investigation of formal logic.
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This suggests at least that the awareness of the existence of concepts with "fuzzy" characteristics, in one form or another, has a very long history in human thought. Quite a few logicians and philosophers have also tried to analyze the characteristics of fuzzy concepts as a recognized species, sometimes with the aid of some kind of many-valued logic or substructural logic. An early attempt in the post-WW2 era to create a theory of sets where set membership is a matter of degree was made by Abraham Kaplan and Hermann Schott in 1951. They intended to apply the idea to empirical research.
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The Italian mathematician Mario Pieri (1860-1913) took a different approach and considered a system in which there were only two primitive notions, that of point and of motion. Pasch had used four primitives and Peano had reduced this to three, but both of these approaches relied on some concept of betweeness which Pieri replaced by his formulation of motion. In 1905 Pieri gave the first axiomatic treatment of complex projective geometry which did not start by building real projective geometry. Pieri was a member of a group of Italian geometers and logicians that Peano had gathered around himself in Turin.
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Thus, swapping left for right in a sequent corresponds to negating all of the constituent formulae. This means that a symmetry such as De Morgan's laws, which manifests itself as logical negation on the semantic level, translates directly into a left-right symmetry of sequents — and indeed, the inference rules in sequent calculus for dealing with conjunction (∧) are mirror images of those dealing with disjunction (∨). Many logicians feel that this symmetric presentation offers a deeper insight in the structure of the logic than other styles of proof system, where the classical duality of negation is not as apparent in the rules.
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Graham Priest and other logicians, including J. C. Beall and Bradley Armour-Garb, have proposed that the liar sentence should be considered to be both true and false, a point of view known as dialetheism. Dialetheism is the view that there are true contradictions. Dialetheism raises its own problems. Chief among these is that since dialetheism recognizes the liar paradox, an intrinsic contradiction, as being true, it must discard the long- recognized principle of explosion, which asserts that any proposition can be deduced from a contradiction, unless the dialetheist is willing to accept trivialism – the view that all propositions are true.
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Consequently, many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan—completely independent of Leibniz. Just as propositional logic can be considered an advancement from the earlier syllogistic logic, Gottlob Frege's predicate logic can be also considered an advancement from the earlier propositional logic. One author describes predicate logic as combining "the distinctive features of syllogistic logic and propositional logic." Consequently, predicate logic ushered in a new era in logic's history; however, advances in propositional logic were still made after Frege, including natural deduction, truth trees and truth tables.
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His agreement with any system was only conditional and formal; he always had innumerable objections to every thing he seemed to acquiesce in. Hence his contention with the predicaments of Aristotle; his attempted refutation of the term universal; and the contrasts he instituted between the whole system of the Stagirite, and the philosophical opinions of Plato, Democritus, Epicurus, and many other distinguished men of antiquity. Derodon took great delight in discussions on the nature of genus and species, and on those curious and puzzling questions which go under the name of the Cross of Logicians. The following are some of the debatable points.
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Here cup as an utterance signifies a cup as an object, but cup as a term of the language English is being used to supposit for the wine contained in the cup. Medieval logicians divided supposition into many different kinds, and the jargons for the different kinds, and their relations and what they all mean get complex, and differ greatly from logician to logician.Marcia L. Colish (1976) Medieval Foundations of the Western intellectual Tradition, pages 275,6, Yale University Press Paul Spade's webpage has a series of helpful diagrams here. The most important division is probably between material, simple, personal, and improper supposition.
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Although the liar paradox was well known in antiquity, interest seems to have lapsed until the twelfth century, when it appears to have been reinvented independently of ancient authors. Medieval interest may have been inspired by a passage in the Sophistical Refutations of Aristotle. Although the Sophistical Refutations are consistently cited by medieval logicians from the earliest insolubilia literature, medieval studies of insolubilia go well beyond Aristotle. Other ancient sources which could suggest the liar paradox, including Saint Augustine, Cicero, and the quotation of Epimenides appearing in the Epistle to Titus, were not cited in discussions of insolubilia.
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Thus, a formal proof is less intuitive, and less susceptible to logical errors.C. Hales, Thomas "Formal Proof", University of Pittsburgh. Retrieved on 2010-10-19 Some consider the Cornell Summer meeting of 1957, which brought together many logicians and computer scientists, as the origin of automated reasoning, or automated deduction."Automated Deduction (AD)", [The Nature of PRL Project]. Retrieved on 2010-10-19 Others say that it began before that with the 1955 Logic Theorist program of Newell, Shaw and Simon, or with Martin Davis’ 1954 implementation of Presburger's decision procedure (which proved that the sum of two even numbers is even).
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Chairs were renamed to departments and the Faculty of Philosophical Studies were forcibly joined with the Faculty of Sociology, within which the Institute of Philosophy was established. In the post-war period the tradition of Lvov-Warsaw School was also continued. To the most important philosophers and logicians of the post-war period connected with the University of Warsaw belong Kazimierz Ajdukiewicz, Zdzisław Augustynek, Bronisław Baczko, Marek Fritzhand, Henryk Jankowski, Andrzej Kasia, Leszek Kołakowski, Władysław Krajewski, Tadeusz Kroński, Jan Legowicz, Stefan Morawski, Elżbieta Pietruska-Madej, Marian Przełęcki, Adam Schaff, Marek Siemek, Adam Sikora, Roman Suszko, Klemens Szaniawski and Bogusław Wolniewicz.
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By design, the letter shape of each X-stem Logic Alphabet symbol visually embodies and displays its individual underlying logic truth table. In other words, after the simple and exact truth table code in the deep structure of XLA has been learned, operations performed on the letter shape symbols are equivalent to logical operations acting on highly abbreviated sets of mini truth tables. Consequently, those using XLA never have the need to interrupt their calculations to check rows and columns of laid out truth tables. This basic and central advantage of XLA over PWR is often not fully recognized, even by practiced logicians.
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Between 1955 and 1959 she held temporary positions teaching elementary math, ancient philosophy and logic, and was a research associate, in the University of Chicago, Indiana University Northwest in Gary, and Notre Dame University. In 1959 she returned to Cambridge, Massachusetts and after that to Princeton, New Jersey. In the following years she earned her living from grants and fellowships which were given to her mostly for her work on translations of Polish and Russian logicians. When not supported by grants Rand operated on private loans and other financial assistance, freelance translation work, or sporadic temporary employment.
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Presented to the Society 13 December 1912. Sheffer introduced what is now known as the Sheffer stroke in 1913; it became well known only after its use in the 1925 edition of Whitehead and Russell's Principia Mathematica. Sheffer's discovery won great praise from Bertrand Russell, who used it extensively to simplify his own logic, in the second edition of his Principia Mathematica. Because of this comment, Sheffer was something of a mystery man to logicians, especially because Sheffer, who published little in his career, never published the details of this method, only describing it in mimeographed notes and in a brief published abstract.
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Naive set theory (the axiom schema of unrestricted comprehension and the axiom of extensionality) is inconsistent due to Russell's paradox. In early formalizations of sets, mathematicians and logicians have avoided that contradiction by replacing the axiom schema of comprehension with the much weaker axiom schema of separation. However, this step alone takes one to theories of sets which are considered too weak. So some of the power of comprehension was added back via the other existence axioms of ZF set theory (pairing, union, powerset, replacement, and infinity) which may be regarded as special cases of comprehension.
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Since the late 20th century, European and American logicians have attempted to provide mathematical foundations for dialectical logic or argument. There had been pre-formal and partially-formal treatises on argument and dialectic, from authors such as Stephen Toulmin (The Uses of Argument), Nicholas Rescher (Dialectics), and van Eemeren and Grootendorst (pragma-dialectics). One can include the communities of informal logic and paraconsistent logic. However, building on theories of defeasible reasoning (see John L. Pollock), systems have been built that define well-formedness of arguments, rules governing the process of introducing arguments based on fixed assumptions, and rules for shifting burden.
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Focusing on IF- THEN statements (what logicians call modus ponens) still gave developers a very powerful general mechanism to represent logic, but one that could be used efficiently with computational resources. What is more there is some psychological research that indicates humans also tend to favor IF-THEN representations when storing complex knowledge. A simple example of modus ponens often used in introductory logic books is "If you are human then you are mortal". This can be represented in pseudocode as: Rule1: Human(x) => Mortal(x) A trivial example of how this rule would be used in an inference engine is as follows.
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Logicians analyze how analogical reasoning is used in arguments from analogy. An analogy can be stated using is to and as when representing the analogous relationship between two pairs of expressions, for example, "Smile is to mouth, as wink is to eye." In the field of mathematics and logic, this can be formalized with colon notation to represent the relationships, using single colon for ratio, and double colon for equality. In the field of testing, the colon notation of ratios and equality is often borrowed, so that the example above might be rendered, "Smile : mouth :: wink : eye" and pronounced the same way.
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Platon Poretsky Platon Sergeevich Poretsky (; October 3, 1846 in Elisavetgrad – August 9, 1907 in Chernihiv Governorate) was a noted Russian astronomer, mathematician, and logician. Graduated from Kharkov University, he worked in Astrakhan and Pulkovo. Later, as an astronomer at Kazan University, following the advice of his older colleague Professor of Mathematics A.V. Vasiliev at Kazan University (father of Nicolai A. Vasiliev) to learn the works of George Boole, Poretsky developed "logical calculus" and through specific "logical equations" applied it to the theory of probability. Thus, he extended and augmented the works of logicians and mathematicians George Boole, William Stanley Jevons and Ernst Schröder.
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Psychologists (and economists) have classified and described a sizeable catalogue of biases which recur frequently in human thought. The availability heuristic, for example, is the tendency to overestimate the importance of something which happens to come readily to mind. Elements of behaviorism and cognitive psychology were synthesized to form cognitive behavioral therapy, a form of psychotherapy modified from techniques developed by American psychologist Albert Ellis and American psychiatrist Aaron T. Beck. On a broader level, cognitive science is an interdisciplinary enterprise of cognitive psychologists, cognitive neuroscientists, researchers in artificial intelligence, linguists, human–computer interaction, computational neuroscience, logicians and social scientists.
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Among Wolniewicz's direct students are his doctoral candidates – Zbigniew Musiał, Ulrich Schrade, Beata Witkowska- Maksimczuk; his graduate students – Agnieszka Maria Nogal, Paweł Okołowski, Klaudiusz Suczyński; participants of Wolniewicz's seminars of many years – Jan Zubelewicz, Jędrzej Stanisławek. These people continue and develop Wolniewicz's work and his style of doing philosophy. Left under a significant influence of Wolniewicz were not only his closest students but also other key Polish philosophers and logicians. Among them Roman Suszko, whose papers on logic were made under a significant influence of the philosophical interpretation of Wittgenstein made by Wolniewicz, especially included in the paper "Things and facts".
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Logicians in the early 20th century aimed to solve the problem of foundations, such as, "What is the true base of mathematics?" The program was to be able to rewrite all mathematics using an entirely syntactical language without semantics. In the words of David Hilbert (referring to geometry), "it does not matter if we call the things chairs, tables and beer mugs or points, lines and planes." The stress on finiteness came from the idea that human mathematical thought is based on a finite number of principles and all the reasonings follow essentially one rule: the modus ponens.
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The medieval era Jain logicians Akalanka and Vidyananda, who were likely contemporaries of Adi Shankara, acknowledged many issues with anekantavada in their texts. For example, Akalanka in his Pramanasamgraha acknowledges seven problems when anekantavada is applied to develop a comprehensive and consistent philosophy: dubiety, contradiction, lack of conformity of bases (), joint fault, infinite regress, intermixture and absence. Vidyananda acknowledged six of those in the Akalanka list, adding the problem of vyatikara (cross breeding in ideas) and apratipatti (incomprehensibility). Prabhācandra, who probably lived in the 11th-century, and several other later Jain scholars accepted many of these identified issues in anekantavada application.
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She was very isolated and lonely at Harvard, with few friends and, initially, no other students even willing to sit next to her in her classes. The nearest restroom to her classes was in a different building, and one of the few buildings with air conditioning in the summers was off-limits to women, even when she was assigned as an instructor to a class in that building. Because there were no logicians at Harvard at that time, she spent five years of her time as a visiting student at the University of California, Berkeley. Her doctoral dissertation was Computable Functions.
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When the book appeared it received many negative reviews mostly from working logicians and mathematicians, among them Michael Dummett, Paul Bernays, and Georg Kreisel. Today Remarks on the Foundations of Mathematics is read mostly by philosophers sympathetic to Wittgenstein and they tend to adopt a more positive stance.Rodych V, Wittgenstein's Philosophy of Mathematics, SEP Wittgenstein's philosophy of mathematics is exposed chiefly by simple examples on which further skeptical comments are made. The text offers an extended analysis of the concept of mathematical proof and an exploration of Wittgenstein's contention that philosophical considerations introduce false problems in mathematics.
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An inference is not true or false, but valid or invalid. However, there is a connection between implication and inference, as follows: if the implication 'if p then q' is true, the inference 'p therefore q' is valid. This was given an apparently paradoxical formulation by Philo, who said that the implication 'if it is day, it is night' is true only at night, so the inference 'it is day, therefore it is night' is valid in the night, but not in the day. The theory of inference (or consequences) was systematically developed in medieval times by logicians such as William of Ockham and Walter Burley.
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What happened to it after this time was the result of the activities of logicians much more gifted than Ghazâlî. This period has tentatively been called the Golden Age of Arabic philosophy (Gutas 2002). It is in this period, and especially in the thirteenth century, that the major changes in the coverage and structure of Avicennan logic were introduced; these changes were mainly introduced in free-standing treatises on logic. It has been observed that the thirteenth century was the time that “doing logic in Arabic was thoroughly disconnected from textual exegesis, perhaps more so than at any time before or since” (El-Rouayheb 2010b: 48–49).
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266 Modern logic is fundamentally a calculus whose rules of operation are determined only by the shape and not by the meaning of the symbols it employs, as in mathematics. Many logicians were impressed by the "success" of mathematics, in that there had been no prolonged dispute about any truly mathematical result. C.S. Peirce notedPeirce 1896 that even though a mistake in the evaluation of a definite integral by Laplace led to an error concerning the moon's orbit that persisted for nearly 50 years, the mistake, once spotted, was corrected without any serious dispute. Peirce contrasted this with the disputation and uncertainty surrounding traditional logic, and especially reasoning in metaphysics.
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The texts include his commentaries on the Mulamadhyamakakarika or Fundamental Stanzas on Wisdom by Nagarjuna; the Introduction to the Middle Way (Sanskrit: Madhyamakāvatāra) of Chandrakirti; the Quintessence of all Courses of Ultimate Wisdom (Jnanasarasamuccaya) of Aryadeva; commentaries on the major works of the Indian Buddhist logicians Dharmakirti and Dignaga; commentaries on the Five Treatises of Maitreya most notably, the Abhisamayalamkara; commentaries on several works of Vasubandhu including the Abhidharmakosha. Mipham's commentary on the ninth chapter of Shantideva's Bodhicaryavatara, the Shertik Norbu Ketaka (),Kapstein, Matthew T. (2000). 'We Are All Gzhan stong pas: Reflections on The Reflexive Nature of Awareness: A Tibetan Madhyamaka Defence. By Paul Williams.
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Afterwards, Solok used the incident as evidence of his viewpoint that Vulcans are superior to humans. Solok published 5 papers on the wrestling match during their time at academy, and after they graduated Solok published over a dozen papers comparing Vulcans and humans, each beginning with an analysis of the wrestling match, and Sisko refused to lose at his favorite sport to Solok. Sisko makes Kasidy promise to keep this between them, but she immediately tells the truth to the whole team, making them understand just how much this means to Sisko. When the game is played, the Logicians (Solok's team) immediately build up a good score.
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The move to view units in natural language (e.g. English) as formal symbols was initiated by Noam Chomsky (it was this work that resulted in the Chomsky hierarchy in formal languages). The generative grammar model looked upon syntax as autonomous from semantics. Building on these models, the logician Richard Montague proposed that semantics could also be constructed on top of the formal structure: :There is in my opinion no important theoretical difference between natural languages and the artificial languages of logicians; indeed, I consider it possible to comprehend the syntax and semantics of both kinds of language within a single natural and mathematically precise theory.
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Early forms of analogical reasoning, inductive reasoning and categorical syllogism were introduced in Fiqh (Islamic jurisprudence), Sharia and Kalam (Islamic theology) from the 7th century with the process of Qiyas, before the Arabic translations of Aristotle's works. Later, during the Islamic Golden Age, there was debate among Islamic philosophers, logicians and theologians over whether the term Qiyas refers to analogical reasoning, inductive reasoning or categorical syllogism. Some Islamic scholars argued that Qiyas refers to inductive reasoning. Ibn Hazm (994–1064) disagreed, arguing that Qiyas does not refer to inductive reasoning but to categorical syllogistic reasoning in a real sense and analogical reasoning in a metaphorical sense.
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He was one of the first English-speaking logicians to appreciate the nature and scope of the logical work of Charles Sanders Peirce, and the distinction between de dicto and de re in modal logic. Prior taught and researched modal logic before Kripke proposed his possible worlds semantics for it, at a time when modality and intentionality commanded little interest in the English speaking world, and had even come under sharp attack by Willard Van Orman Quine. He is now said to be the precursor of hybrid logic. Undertaking (in one section of his book Past, Present, and Future (1967)) the attempt to combine binary (e.g.
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The name of Logic Lane was adopted by the 17th century, owing to the presence of a school of logicians at the northern end of the lane. A medieval street used to run across Logic Lane as an extension of the current Kybald Street to the west, but was closed in 1448. In 1904, a covered bridge at the High Street end of the lane was built to link the older part of the college with the then new Durham Buildings. The lane was officially a public bridleway, and the city council opposed the scheme, but the court judgement was in favour of the college.
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The participants tried to develop a so-called "genetically meaningful" logic – an alternative to both semi-official dialectical logic and formal logic. The activity of the circle took place against the backdrop of the revival of the atmosphere at the philosophical faculty after Stalin's death. At the beginning of 1954, a discussion was held on "Disagreements on Logic Issues", which divided "dialecticians", formal logicians and "heretics" from the circle – the so- called "easel painters". In another discussion, Zinoviev said a well-known phrase that "earlier bourgeois philosophers explained the world, and now Soviet philosophers do not do this", which caused the applause of the audience.
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He lectured at the University of Vienna in the years 1894–95. In 1895 was appointed professor at Lwów (Lemberg in Austrian Galicia, now Lviv in the Ukraine). An outstanding lecturer, he was also a rector of the Lwów University during World War I. There Twardowski soon established the Lwów–Warsaw school of logic and became the "father of Polish logic", beginning the tradition of analytic philosophy in Poland. Among his students were the logicians Stanisław Leśniewski, Jan Łukasiewicz and Tadeusz Czeżowski, the historian of philosophy Władysław Tatarkiewicz, the phenomenologist and aesthetician Roman Ingarden, as well as philosophers close to the Vienna Circle such as Tadeusz Kotarbiński and Kazimierz Ajdukiewicz.
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Logic is concerned with the patterns in reason that can help tell us if a proposition is true or not. Logicians use formal languages to express the truths which they are concerned with, and as such there is only truth under some interpretation or truth within some logical system. A logical truth (also called an analytic truth or a necessary truth) is a statement which is true in all possible worldsLudwig Wittgenstein, Tractatus Logico-Philosophicus. or under all possible interpretations, as contrasted to a fact (also called a synthetic claim or a contingency) which is only true in this world as it has historically unfolded.
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Disarming and placing Klieg and Kaftan in the weapon testing room whilst they wait for the rocket to be repaired, the group is attacked by a swarm of Cybermats, which the Doctor incapacitates with electrical currents. After repairing a cybergun on the dummy, the Logicians return and open the hatch, believing that they can still forge their alliance with the Cybermen. With their energy levels running low, the Cybermen return to their tombs whilst the Cyber-Controller and a partially converted Toberman meet with the group. Taking him to the revitalizing chamber, the Doctor attempts to sabotage the process, only for the Controller to escape and turn on the group.
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Thus, from a modern point of view, it often makes sense to talk about 'the' opposition of a claim, rather than insisting as older logicians did that a claim has several different opposites, which are in different kinds of opposition with the claim. Gottlob Frege's Begriffsschrift also presents a square of oppositions, organised in an almost identical manner to the classical square, showing the contradictories, subalternates and contraries between four formulae constructed from universal quantification, negation and implication. Algirdas Julien Greimas' semiotic square was derived from Aristotle's work. The traditional square of opposition is now often compared with squares based on inner- and outer-negation.
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Abstract approach on how knowledge representation and reasoning allow a problem specific solution (answer) to a given problem (questions) Representing meaning as a graph is one of the two ways that both an AI cognition and a linguistic researcher think about meaning (connectionist view). Logicians utilize a formal representation of meaning to build upon the idea of symbolic representation, whereas description logics describe languages and the meaning of symbols. This contention between 'neat' and 'scruffy' techniques has been discussed since the 1970s. Research has so far identified semantic measures and with that Word-sense disambiguation (WSD) - the differentiation of meaning of words - as the main problem of language understanding.
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The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory, and they are a key reason for the prominence of first-order logic in mathematics. Gödel's incompleteness theorems (Gödel 1931) establish additional limits on first- order axiomatizations.
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The validity of an argument depends upon the meaning or semantics of the sentences that make it up. Aristotle's six Organon, especially De Interpretatione, gives a cursory outline of semantics which the scholastic logicians, particularly in the thirteenth and fourteenth century, developed into a complex and sophisticated theory, called supposition theory. This showed how the truth of simple sentences, expressed schematically, depend on how the terms 'supposit', or stand for, certain extra-linguistic items. For example, in part II of his Summa Logicae, William of Ockham presents a comprehensive account of the necessary and sufficient conditions for the truth of simple sentences, in order to show which arguments are valid and which are not.
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George Boole presented this expansion as his Proposition II, "To expand or develop a function involving any number of logical symbols", in his Laws of Thought (1854),George Boole, An Investigation of the Laws of Thought: On which are Founded the Mathematical Theories of Logic and Probabilities, 1854, p. 72 full text at Google Books and it was "widely applied by Boole and other nineteenth-century logicians".Frank Markham Brown, Boolean Reasoning: The Logic of Boolean Equations, 2nd edition, 2003, p. 42 Claude Shannon mentioned this expansion, among other Boolean identities, in a 1948 paper,Claude Shannon, "The Synthesis of Two-Terminal Switching Circuits", Bell System Technical Journal 28:59–98, full text, p.
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The main proponent of such a theory is Noam Chomsky, the originator of the generative theory of grammar, who has defined language as the construction of sentences that can be generated using transformational grammars. Chomsky considers these rules to be an innate feature of the human mind and to constitute the rudiments of what language is. By way of contrast, such transformational grammars are also commonly used in formal logic, in formal linguistics, and in applied computational linguistics. In the philosophy of language, the view of linguistic meaning as residing in the logical relations between propositions and reality was developed by philosophers such as Alfred Tarski, Bertrand Russell, and other formal logicians.
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Some psychologists and logicians argue that fuzzy concepts are a necessary consequence of the reality that any kind of distinction we might like to draw has limits of application. At a certain level of generality, a distinction works fine. But if we pursued its application in a very exact and rigorous manner, or overextend its application, it appears that the distinction simply does not apply in some areas or contexts, or that we cannot fully specify how it should be drawn. An analogy might be, that zooming a telescope, camera, or microscope in and out, reveals that a pattern which is sharply focused at a certain distance becomes blurry at another distance, or disappears altogether.
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Common use of this sort of approach (combining words and numbers in programming), has led some logicians to regard fuzzy logic merely as an extension of Boolean logic (a two-valued logic or binary logic is simply replaced with a many-valued logic). However, Boolean concepts have a logical structure which differs from fuzzy concepts. An important feature in Boolean logic is, that an element of a set can also belong to any number of other sets; even so, the element either does, or does not belong to a set (or sets). By contrast, whether an element belongs to a fuzzy set is a matter of degree, and not always a definite yes-or-no question.
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But the conclusion, considered by itself and with the possible authors not limited to just Shakespeare and Hobbes, is dubious, because if Shakespeare is ruled out as _Hamlet_ 's author, there are many more plausible alternatives than Hobbes. The general form of McGee-type counterexamples to modus ponens is simply P, P \rightarrow (Q \rightarrow R), therefore Q \rightarrow R; it is not essential that P be a disjunction, as in the example given. That these kinds of cases constitute failures of modus ponens remains a minority view among logicians, but opinions vary on how the cases should be disposed of.Sinnott-Armstrong, Moor, and Fogelin (1986). "A Defense of Modus Ponens", The Journal of Philosophy 83, 296–300.
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Fakhr al-Din al-Razi (b. 1149) criticised Aristotle's "first figure" and developed a form of inductive logic, foreshadowing the system of inductive logic developed by John Stuart Mill (1806–1873). Systematic refutations of Greek logic were written by the Illuminationist school, founded by Shahab al-Din Suhrawardi (1155–1191), who developed the idea of "decisive necessity", an important innovation in the history of logical philosophical speculation. Another systematic refutation of Greek logic was written by Ibn Taymiyyah (1263-1328), the Ar-Radd 'ala al-Mantiqiyyin (Refutation of Greek Logicians), where he argued against the usefulness, though not the validity, of the syllogismSee pp. 253-254 of and in favour of inductive reasoning.
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Many techniques are employed by logicians to represent an argument's logical form. A simple example, applied to two of the above illustrations, is the following: Let the letters 'P', 'Q', and 'S' stand, respectively, for the set of men, the set of mortals, and Socrates. Using these symbols, the first argument may be abbreviated as: : All P are Q. : S is a P. : Therefore, S is a Q. Similarly, the second argument becomes: : All P are not Q. : S is a P. : Therefore, S is a Q. An argument is termed formally valid if it has structural self-consistency, i.e. if when the operands between premises are all true, the derived conclusion is always also true.
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Kurt Friedrich Gödel (; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and analytic philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an immense effect upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell,For instance, in their Principia Mathematica (Stanford Encyclopedia of Philosophy edition). Alfred North Whitehead, and David Hilbert were analyzing the use of logic and set theory to understand the foundations of mathematics pioneered by Georg Cantor. Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna.
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A text by Avicenna, founder of Avicennian logic The works of Al-Kindi, Al-Farabi, Avicenna, Al-Ghazali, Averroes and other Muslim logicians were based on Aristotelian logic and were important in communicating the ideas of the ancient world to the medieval West.See e.g. Routledge Encyclopedia of Philosophy Online Version 2.0 , article 'Islamic philosophy' Al-Farabi (Alfarabi) (873–950) was an Aristotelian logician who discussed the topics of future contingents, the number and relation of the categories, the relation between logic and grammar, and non-Aristotelian forms of inference. Al-Farabi also considered the theories of conditional syllogisms and analogical inference, which were part of the Stoic tradition of logic rather than the Aristotelian. [726].
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Leibniz The idea that inference could be represented by a purely mechanical process is found as early as Raymond Llull, who proposed a (somewhat eccentric) method of drawing conclusions by a system of concentric rings. The work of logicians such as the Oxford CalculatorsEdith Sylla (1999), "Oxford Calculators", in The Cambridge Dictionary of Philosophy, Cambridge, Cambridgeshire: Cambridge. led to a method of using letters instead of writing out logical calculations (calculationes) in words, a method used, for instance, in the Logica magna by Paul of Venice. Three hundred years after Llull, the English philosopher and logician Thomas Hobbes suggested that all logic and reasoning could be reduced to the mathematical operations of addition and subtraction.
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Since the early 20th century, certain logicians have proposed logics that deny the validity of the law. Logics known as "paraconsistent" are inconsistency- tolerant logics in that there, from P together with ¬P, it doesn't imply that any proposition follows. Nevertheless, not all paraconsistent logics deny the law of non-contradiction and some such logics even prove it. In several axiomatic derivations of logic,Steven Wolfram, A New Kind Of Science, this is effectively resolved by showing that (P ∨ ¬P) and its negation are constants, and simply defining TRUE as (P ∨ ¬P) and FALSE as ¬(P ∨ ¬P), without taking a position as to the principle of bivalence or the law of excluded middle.
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The Association Computability in Europe (ACiE) is an international organization of mathematicians, logicians, computer scientists, philosophers, theoretical physicists and others interested in new developments in computability and in their underlying significance for the real world. CiE aims to widen understanding and appreciation of the importance of the concepts and techniques of computability theory, and to support the development of a vibrant multi-disciplinary community of researchers focused on computability- related topics. The ACiE positions itself at the interface between applied and fundamental research, prioritising mathematical approaches to computational barriers. The Association Computability in Europe originated as a research network called Computability in Europe (CiE) in 2003, became a conference series in 2005, and the ACiE was formed in 2008.
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The completeness of first-order logic is a corollary of results Skolem proved in the early 1920s and discussed in Skolem (1928), but he failed to note this fact, perhaps because mathematicians and logicians did not become fully aware of completeness as a fundamental metamathematical problem until the 1928 first edition of Hilbert and Ackermann's Principles of Mathematical Logic clearly articulated it. In any event, Kurt Gödel first proved this completeness in 1930. Skolem distrusted the completed infinite and was one of the founders of finitism in mathematics. Skolem (1923) sets out his primitive recursive arithmetic, a very early contribution to the theory of computable functions, as a means of avoiding the so-called paradoxes of the infinite.
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Although the rest of the galaxy presumed them to have died out, they had, in reality, retreated into hibernation in the tombs of Telos to await discovery and reactivation. Sometime around the 25th century, a team of archeologists, led by Professor Parry and financed by the Brotherhood of Logicians, embarked on an expedition to Telos, believing it to be the Cyberman homeworld and hoping to discover artefacts amongst its ruins. No Cryons were encountered during these events and it is not known what the Cryons were doing during the Cybermen's long sleep. Uncovering the entrance to the tombs, Eric Klieg of the Brotherhood brought the Cybermen out of hibernation in an attempt to ally the Brotherhood with them.
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Indexicals appear to represent an exception to, and thus a challenge for, the understanding of natural language as the grammatical coding of logical propositions; they thus "raise interesting technical challenges for logicians seeking to provide formal models of correct reasoning in natural language." They are also studied in relation to fundamental issues in epistemology, self-consciousness, and metaphysics, for example asking whether indexical facts are facts that do not follow from the physical facts, and thus also form a link between philosophy of language and philosophy of mind. The American logician David Kaplan is regarded as having developed "[b]y far the most influential theory of the meaning and logic of indexicals".
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Further evidence that he lectured in Paris is that those logicians who were influenced by his work also worked there, including Peter of Spain () and Lambert of Auxerre (). He is thought to have become treasurer of Lincoln Cathedral some time in the 1250s. The treasurer was one of the four principal officers of the English cathedrals whose duty was to keep the treasures of the church – the gold and silver vessels, ornaments, relics, jewels, and altar cloths. He would have had a personal residence in the Cathedral close, would have employed a deputy and a large staff, and therefore could be absent as long as he performed those duties that could not be delegated.
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Victor Ivanovich Shestakov (1907–1987) was a Russian/Soviet logician and theoretician of electrical engineering. In 1935 he discovered the possible interpretation of Boolean algebra of logic in electro-mechanical relay circuits. He graduated from Moscow State University (1934) and worked there in the General Physics Department almost until his death. Shestakov proposed a theory of electric switches based on Boolean logic earlier than Claude Shannon (according to certification of Soviet logicians and mathematicians Sofya Yanovskaya, M.G. Gaaze-Rapoport, Roland Dobrushin, Oleg Lupanov, Yu. A. Gastev, Yu. T. Medvedev, and Vladimir Andreevich Uspensky), though Shestakov and Shannon defended Theses the same year (1938) and the first publication of Shestakov's result took place only in 1941 (in Russian).
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The basic ideas of modal logic date back to antiquity. Aristotle developed a modal syllogistic in Book I of his Prior Analytics (chs 8–22), which Theophrastus attempted to improve. There are also passages in Aristotle's work, such as the famous sea-battle argument in De Interpretatione §9, that are now seen as anticipations of the connection of modal logic with potentiality and time. In the Hellenistic period, the logicians Diodorus Cronus, Philo the Dialectician and the Stoic Chrysippus each developed a modal system that accounted for the interdefinability of possibility and necessity, accepted axiom T (see below), and combined elements of modal logic and temporal logic in attempts to solve the notorious Master Argument.Bobzien, S. (1993).
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Stillit was born in London, the youngest of four children to Joseph Stillitz, subsequently anglicising his name by deed poll. Stillit was educated at Highgate School in North London, leaving at 16 after having obtained O-Levels to train as a Chartered Accountant. He thereupon attended the London School of Economics, reading towards a Bachelor's in Economics under logicians Karl Popper and Imre Lakatos – however left aged 25 before completion of his degree as the success of his inventions overtook him. Concurrently, he also worked in his father's company Gor-Ray, at the time Britain's foremost skirt manufacturer, and while there introduced innovative retailing concepts such as Britain's first boutique concession store, operating within House of Fraser.
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Russell's influence is also evident in the work of Alfred J. Ayer, Rudolf Carnap, Alonzo Church, Kurt Gödel, David Kaplan, Saul Kripke, Karl Popper, W. V. Quine, John R. Searle, and a number of other philosophers and logicians. Russell often characterised his moral and political writings as lying outside the scope of philosophy, but Russell's admirers and detractors are often more acquainted with his pronouncements on social and political matters, or what some (e.g., biographer Ray Monk) have called his "journalism," than they are with his technical, philosophical work. There is a marked tendency to conflate these matters, and to judge Russell the philosopher on what he himself would definitely consider to be his non- philosophical opinions.
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In the Posterior Analytics,Posterior Analytics Bk 2 c. 7 he says that the meaning of a made-up name can be known (he gives the example "goat stag") without knowing what he calls the "essential nature" of the thing that the name would denote (if there were such a thing). This led medieval logicians to distinguish between what they called the quid nominis, or the "whatness of the name", and the underlying nature common to all the things it names, which they called the quid rei, or the "whatness of the thing".. Early modern philosophers like Locke used the corresponding English terms "nominal essence" and "real essence". The name "hobbit", for example, is perfectly meaningful.
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Bolzano's posthumously published work Paradoxien des Unendlichen (The Paradoxes of the Infinite) (1851) was greatly admired by many of the eminent logicians who came after him, including Charles Sanders Peirce, Georg Cantor, and Richard Dedekind. Bolzano's main claim to fame, however, is his 1837 Wissenschaftslehre (Theory of Science), a work in four volumes that covered not only philosophy of science in the modern sense but also logic, epistemology and scientific pedagogy. The logical theory that Bolzano developed in this work has come to be acknowledged as ground-breaking. Other works are a four-volume Lehrbuch der Religionswissenschaft (Textbook of the Science of Religion) and the metaphysical work Athanasia, a defense of the immortality of the soul.
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The incompleteness theorem is sometimes thought to have severe consequences for the program of logicism proposed by Gottlob Frege and Bertrand Russell, which aimed to define the natural numbers in terms of logic (Hellman 1981, p. 451–468). Bob Hale and Crispin Wright argue that it is not a problem for logicism because the incompleteness theorems apply equally to first order logic as they do to arithmetic. They argue that only those who believe that the natural numbers are to be defined in terms of first order logic have this problem. Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to David Hilbert's second problem, which asked for a finitary consistency proof for mathematics.
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Although, like Ockham, he refused to construe relations as things distinct from absolute entities, he clearly ascribed them to an act of the soul by which absolute entities are compared and placed in relation to each other. He therefore completely rejected certain propositions Ockham had admitted reasonable, even if he did not construe them in the same way. Albert's voluminous collection of Sophismata (c. 1359) examined various sentences that raise difficulties of interpretation due to the presence of syncategorematic terms such as quantifiers and certain prepositions, which, according to medieval logicians, do not have a proper and determinate signification but rather modify the signification of the other terms in the propositions in which they occur.
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The solution of a major unsolved problem some years later led to a new treatment, The Logic of Provability, published in 1993. The modal-logical treatment of provability helped demonstrate the "intensionality" of Gödel's Second Incompleteness Theorem, meaning that the theorem's correctness depends on the precise formulation of the provability predicate. These conditions were first identified by David Hilbert and Paul Bernays in their Grundlagen der Arithmetik. The unclear status of the Second Theorem was noted for several decades by logicians such as Georg Kreisel and Leon Henkin, who asked whether the formal sentence expressing "This sentence is provable" (as opposed to the Gödel sentence, "This sentence is not provable") was provable and hence true.
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Russell and G. E. Moore broke themselves free from British Idealism which, for nearly 90 years, had dominated British philosophy. Russell would later recall in "My Mental Development" that "with a sense of escaping from prison, we allowed ourselves to think that grass is green, that the sun and stars would exist if no one was aware of them..."—Russell B, (1944) "My Mental Development", in Schilpp, Paul Arthur: The Philosophy of Bertrand Russell, New York: Tudor, 1951, pp. 3–20. He is considered one of the founders of analytic philosophy along with his predecessor Gottlob Frege, colleague G. E. Moore and protégé Ludwig Wittgenstein. He is widely held to be one of the 20th century's premier logicians.
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Wittgenstein wrote about the paradox extensively in his later writings, which brought Moore's paradox the attention it would not have otherwise received. Moore's paradox has been connected to many other well-known logical paradoxes, including, though not limited to, the liar paradox, the knower paradox, the unexpected hanging paradox, and the preface paradox. There is currently no generally accepted explanation of Moore's paradox in the philosophical literature. However, while Moore's paradox remains a philosophical curiosity, Moorean-type sentences are used by logicians, computer scientists, and those working in the artificial intelligence community as examples of cases in which a knowledge, belief, or information system is unsuccessful in updating its fund of knowledge, belief, or information in light of new or novel data.
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Early Islamic law placed importance on formulating standards of argument, which gave rise to a "novel approach to logic" ( manṭiq "speech, eloquence") in Kalam (Islamic scholasticism). However, with the rise of the Mu'tazili philosophers, who highly valued Aristotle's Organon, this approach was displaced by the older ideas from Hellenistic philosophy. The works of al- Farabi, Avicenna, al-Ghazali and other Persian Muslim logicians who often criticized and corrected Aristotelian logic and introduced their own forms of logic, also played a central role in the subsequent development of European logic during the Renaissance. The use of Aristotelian logic in Islamic theology again began to decline from the 10th century, with the rise of Ashʿari theology to the intellectual mainstream, which rejects causal reasoning in favour of clerical authority.
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In 1908, she published the essay "Militant Tactics and Woman's Suffrage" and took part in the second Hyde Park women's suffrage demonstration. She was also opposed to vivisection, writing much on the subject, including "The Sanctuary of Mercy" (1895), "Beyond the Pale" (1896), "The Ethics of Vivisection" (1900), and a play, "The Logicians: An episode in dialogue" (1902), where characters argue opposing views on the issue. Caird was a member of the Theosophical Society from 1904 to 1909. Among her later writings is an illustrated volume of travel essays, Romantic Cities of Provence (1906), and novels: The Stones of Sacrifice (1915), showing harmful effects of self-sacrifice on women, and The Great Wave (1931), a work of social science fiction attacking the racism of negative eugenics.
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At about the same time (1912) that Russell and Whitehead were finishing the last volume of their Principia Mathematica, and the publishing of Russell's "The Problems of Philosophy" at least two logicians (Louis Couturat, Christine Ladd-Franklin) were asserting that two "laws" (principles) of contradiction" and "excluded middle" are necessary to specify "contradictories"; Ladd-Franklin renamed these the principles of exclusion and exhaustion. The following appears as a footnote on page 23 of Couturat 1914: :"As Mrs. LADD·FRANKLlN has truly remarked (BALDWIN, Dictionary of Philosophy and Psychology, article "Laws of Thought"), the principle of contradiction is not sufficient to define contradictories; the principle of excluded middle must be added which equally deserves the name of principle of contradiction. This is why Mrs.
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Psychologism in the philosophy of mathematics is the position that mathematical concepts and/or truths are grounded in, derived from or explained by psychological facts (or laws). John Stuart Mill seems to have been an advocate of a type of logical psychologism, as were many 19th- century German logicians such as Sigwart and Erdmann as well as a number of psychologists, past and present: for example, Gustave Le Bon. Psychologism was famously criticized by Frege in his The Foundations of Arithmetic, and many of his works and essays, including his review of Husserl's Philosophy of Arithmetic. Edmund Husserl, in the first volume of his Logical Investigations, called "The Prolegomena of Pure Logic", criticized psychologism thoroughly and sought to distance himself from it.
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John Buridan (c. 1300 – 1361), whom some consider the foremost logician of the later Middle Ages, contributed two significant works: Treatise on Consequence and Summulae de Dialectica, in which he discussed the concept of the syllogism, its components and distinctions, and ways to use the tool to expand its logical capability. For 200 years after Buridan's discussions, little was said about syllogistic logic. Historians of logic have assessed that the primary changes in the post-Middle Age era were changes in respect to the public's awareness of original sources, a lessening of appreciation for the logic's sophistication and complexity, and an increase in logical ignorance—so that logicians of the early 20th century came to view the whole system as ridiculous.
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In his opening address, "The Four-Fold Art of Avoiding Questions", Paul Weiss spoke of the need for a society that would reinvigorate philosophic inquiry. He denounced "parochialism," referring to those who insisted upon "some one method, say that of pragmatism, instrumentalism, idealism, analysis, linguistics or logistics, and denied the importance of meaningfulness of anything which lies beyond its scope or power," as well as those who confined their studies to only some historic era. Early in the history of the society, there was some dispute about whether certain schools of thought should be included in the program. By the second meeting there was controversy regarding papers by logicians, a controversy possibly fueled by the dominance of positivism in that decade.
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In 1996-2000 the department was joined by Dimiter Dobrev, Jordan Zashev and Dimitar Guelev. From 1971 to 1989 the department was merged with the corresponding division of the Faculty of Mathematics and Informatics at Sofia University, with Dimiter Skordev heading the integrated structure since 1971. In 1989 the institutional relationship with Sofia University was severed, and the department resumed as a division of the Institute of Mathematics and Informatics, headed since then by Lyubomir Ivanov. The logicians Bogdan Dyankov, Hristo Smolenov, Veselin Petrov and Marion Mircheva stayed with the department for various periods of time, all of them coming from the Institute of Philosophy at the Bulgarian Academy of Sciences once the latter was dissolved on account of the dissident activities of its members in 1989.
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In early Islamic philosophy, logic played an important role. Sharia (Islamic law) placed importance on formulating standards of argument, which gave rise to a novel approach to logic in Kalam, but this approach was later displaced by ideas from Greek philosophy and Hellenistic philosophy with the rise of the Mu'tazili philosophers, who highly valued Aristotle's Organon. The works of Hellenistic- influenced Islamic philosophers were crucial in the reception of Aristotelian logic in medieval Europe, along with the commentaries on the Organon by Averroes. The works of al-Farabi, Avicenna, al-Ghazali and other Muslim logicians who often criticized and corrected Aristotelian logic and introduced their own forms of logic, also played a central role in the subsequent development of European logic during the Renaissance.
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Paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers. "This sentence is false" is an example of the well-known liar paradox: it is a sentence which cannot be consistently interpreted as either true or false, because if it is known to be false, then it can be inferred that it must be true, and if it is known to be true, then it can be inferred that it must be false. Russell's paradox, which shows that the notion of the set of all those sets that do not contain themselves leads to a contradiction, was instrumental in the development of modern logic and set theory. Thought-experiments can also yield interesting paradoxes.
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One reason that this particular aspect of intuitionistic logic is so valuable is that it enables practitioners to utilize a wide range of computerized tools, known as proof assistants. These tools assist their users in the verification (and generation) of large-scale proofs, whose size usually precludes the usual human-based checking that goes into publishing and reviewing a mathematical proof. As such, the use of proof assistants (such as Agda or Coq) is enabling modern mathematicians and logicians to develop and prove extremely complex systems, beyond those which are feasible to create and check solely by hand. One example of a proof which was impossible to formally verify without algorithm is the famous proof of the four color theorem.
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In early Islamic philosophy, logic played an important role. Islamic law placed importance on formulating standards of argument, which gave rise to a novel approach to logic in Kalam, but this approach was later displaced by ideas from Greek philosophy and Hellenistic philosophy with the rise of the Mu'tazili philosophers, who highly valued Aristotle's Organon. The works of Hellenistic-influenced Islamic philosophers were crucial in the reception of Aristotelian logic in medieval Europe, along with the commentaries on the Organon by Averroes. The works of al-Farabi, Avicenna, al-Ghazali and other Muslim logicians who often criticized and corrected Aristotelian logic and introduced their own forms of logic, also played a central role in the subsequent development of European logic during the Renaissance.
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While he did publish a fair body of work (Leśniewski, 1992, is his collected works in English translation), some of it in German, the leading language for mathematics of his day, his writings had limited impact because of their enigmatic style and highly idiosyncratic notation. Leśniewski was also a radical nominalist: he rejected axiomatic set theory at a time when that theory was in full flower. He pointed to Russell's paradox and the like in support of his rejection, and devised his three formal systems as a concrete alternative to set theory. Even though Alfred Tarski was his sole doctoral pupil, Leśniewski nevertheless strongly influenced an entire generation of Polish logicians and mathematicians via his teaching at the University of Warsaw.
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The monograph "The Philosophical Problems of Multivalued Logic" (1960), soon translated into English, was a significant event in Soviet philosophy, although it had flaws. Classical work became one of the world's first monographs on multi-valued logic and the first in the Soviet bloc. In general, the work of Zinoviev corresponded to the level of scientific achievements in the field of non-classical logic of the time, highly valued by such logicians as Kazimir Aydukevich, Jozef Bohensky, Georg von Wright, but did not attract much attention in the West. Zinoviev gave priority to formal methods over formal calculi, which alienated his work from the main directions and trends of the logic and methodology of science of the second half of the 20th century.
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Hence my cognition > is supposed to confirm itself, which is far short of being sufficient for > truth. For since the object is outside me, the cognition in me, all I can > ever pass judgement on is whether my cognition of the object agrees with my > cognition of the object. The ancients called such a circle in explanation a > diallelon. And actually the logicians were always reproached with this > mistake by the sceptics, who observed that with this definition of truth it > is just as when someone makes a statement before a court and in doing so > appeals to a witness with whom no one is acquainted, but who wants to > establish his credibility by maintaining that the one who called him as > witness is an honest man.
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In 1937 Shannon went on to write a master's thesis, at the Massachusetts Institute of Technology, in which he showed how Boolean algebra could optimise the design of systems of electromechanical relays then used in telephone routing switches. He also proved that circuits with relays could solve Boolean algebra problems. Employing the properties of electrical switches to process logic is the basic concept that underlies all modern electronic digital computers. Victor Shestakov at Moscow State University (1907–1987) proposed a theory of electric switches based on Boolean logic even earlier than Claude Shannon in 1935 on the testimony of Soviet logicians and mathematicians Sofya Yanovskaya, Gaaze- Rapoport, Roland Dobrushin, Lupanov, Medvedev and Uspensky, though they presented their academic theses in the same year, 1938.
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According to Hemachandra, Anavastha is a Dosha, a defect or fault along with virodha, vaiyadhikarana, samkara, samsaya, vyatikara, apratipatti and abhava. It is also one of the dialectical principles applied alongside atmasraya, anyonyasraya, cakraka, atiprasanga, ubhayatahspasa and the like employed by logicians from very early times. Sriharsa explains that dialectical reasoning, which has its foundation in pervasion, can lead to contradiction when the reasoning becomes fallacious, it is the limit of doubt; and since differing unwanted contrary options create new doubts difficult to resolve which lead to anavastha or infinite regress and there is the absence of finality. The argument that contradiction cannot block an infinite regress is rejected; it is the doubter's own behaviour that process the lie to the doubt, that blocks it (pratibandhaka).
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The Indian grammarian-philosopher Bhartrhari (late fifth century AD) dealt with paradoxes such as the liar in a section of one of the chapters of his magnum opus the Vākyapadīya. Although chronologically he precedes all modern treatments of the problem of the liar paradox, it has only very recently become possible for those who cannot read the original Sanskrit sources to confront his views and analyses with those of modern logicians and philosophers because sufficiently reliable editions and translations of his work have only started becoming available since the second half of the 20th century. Bhartrhari's solution fits into his general approach to language, thought and reality, which has been characterized by some as "relativistic", "non-committal" or "perspectivistic".Jan E. M. Houben, "Bhartrhari's Perspectivism (1)" in Beyond Orientalism ed.
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Sophia is an academic journal devoted to professional pursuits in philosophy, metaphysics, religion and moral thinking, founded in 1962 by Max Charlesworth and Graeme de Graaf. From 2001 Sophia was published by Ashgate Publishing in collaboration with the Australasian Society for Philosophy of Religion and Theology, Australasian Association of Philosophy, Australasian Society for Asian and Comparative Philosophy, The University of Melbourne and Deakin University. The journal has since moved to Springer in Dordrecht-Berlin, with its editorial office split between The University of Melbourne (School of Historical and Philosophical Studies; Hells Logicians) in Australia, Singapore (Philosophy, National University of Singapore), and both coasts of the United States (University of California and Harvard University). The Editors-in-Chief are Purushottama Bilimoria, PhD - also the Managing Executive, Saranindranath Tagore PhD, & Patrick Hutchings, Esquire, Oxon.
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In 1946, he was named Research Assistant at the French National Science Research Centre, a post he was obliged to leave ten years later because of the inability of the Centre to decide in which Scientific Section his work belonged! The next ten or fifteen years were those of greatest acceptance of his work by the public and other thinkers, but unfortunately not by mainstream logicians and philosophers. His Trois Matières, published in 1960 was a bestseller, and people began calling Lupasco the Descartes, the Leibniz, the Hegel of the 20th Century, a new Claude Bernard, a new Bergson, etc. He continued to publish books in the 70s and 80s, the last being L’Homme et ses Trois Ethiques in 1986, two years before his death on October 7, 1988 in Paris.
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Before the Middle Ages there was a logical debate among Islamic logicians, philosophers and theologians over whether the term qiyas refers to analogical reasoning, inductive reasoning or categorical syllogism. Some Islamic scholars argued that qiyas refers to inductive reasoning, which Ibn Hazm (994-1064) disagreed with, arguing that qiyas does not refer to inductive reasoning, but refers to categorical syllogism in a real sense and analogical reasoning in a metaphorical sense. On the other hand, al-Ghazali (1058–1111) and Ibn Qudāmah al-Maqdīsī (1147-1223) argued that qiyas refers to analogical reasoning in a real sense and categorical syllogism in a metaphorical sense. Other Islamic scholars at the time, however, argued that the term qiyas refers to both analogical reasoning and categorical syllogism in a real sense.
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Many of Corcoran's articles and reviews are co-authored and many of his single-author publications acknowledge involvement of colleagues and students. Corcoran emphasizes the intensely and essentially personal nature of all genuine knowledge including logical knowledge. Nevertheless, he also stresses the importance of communities of knowers and how much each person can benefit in the personal search for truth from critical cooperation with other objective researchers. For over 40 years he was the leader of the "Buffalo Syllogistic Group"—a community of philosophers, historians, linguists, logicians, and mathematicians dedicated to the study of the origin of logic. The achievements of this community are sketched in his 2009 paper "Aristotle's Logic at the University at Buffalo's Department of Philosophy", Ideas y Valores: Revista Colombiana de Filosofía 140 (August 2009) 99–117.
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After appearing for the first time in the TOS episode, "Is There in Truth No Beauty?", it appeared in Spock's quarters in Star Trek II: The Wrath of Khan, Star Trek III: The Search for Spock, and Star Trek VI: The Undiscovered Country. In the series Star Trek: Enterprise, T'Pol is given an IDIC pendant from her mother T'Les, she holds an IDIC pendant in "Terra Prime" while she is in mourning for her dying child, and in the episode "The Andorian Incident" the IDIC symbol appears on small playing pieces that are being used to construct a map of the P'Jem catacombs. In the DS9 episode "Take Me Out to the Holosuite", Captain Solok and his Vulcan team, the Logicians, wear ball caps featuring the IDIC symbol.
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Long-term participation in philosophical "gatherings..., in which he spoke with negative views on certain issues of the theory of Marxism-Leninism" (Committee for State Security analytical note) and contacts with American logicians in 1960, according to the Committee for State Security who worked for American intelligence, had their effect. The Organs confined themselves to a conversation (Zinoviev insisted that communication with the Americans had exclusively professional goals), which ended in a curiosity: having learned that he was renting a room, he was given a one-room apartment on Vavilova Street. In the early 1970s, having made an exchange, the Zinovievs moved into a four-room apartment, he had his own office. Later, Zinoviev remarked: "The improvement of living conditions played a huge role in the growth of opposition and rebellious attitudes in the country".
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The Logica Universalis Association, an informal meta-association promoting logic closely linked to Jean-Yves Béziau, promoted the celebration of World Logic Day 2019 by encouraging logicians worldwide to organise independent events on 14 January 2019. Approximately sixty such events were organised in 33 different countries. The success of this informal first World Logic Day formed part of the deliberations of the 40th UNESCO General Conference in November 2019 which led to the formal proclamation by UNESCO. On the first World Logic Day after the UNESCO proclamation, the Director-General of UNESCO, Audrey Azoulay, issued a statement highlighting the importance of logic: > In the twenty-first century – indeed, now more than ever – the discipline of > logic is a particularly timely one, utterly vital to our societies and > economies.
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Fig. 1 – A diagram of A, B, and A \rightarrow B. The statement "The probability that if A, then B, is 20%" means (put intuitively) that event B may be expected to occur in 20% of the outcomes where event A occurs. The standard formal expression of this is P(B\mid A)=0.20, where the conditional probability P(B\mid A) equals, by definition, P(A \cap B)/P(A). Beginning in the 1960s, several philosophical logicians—most notably Ernest Adams and Robert Stalnaker—floated the idea that one might also write P(A \rightarrow B) = 0.20, where A \rightarrow B is the conditional event "If A, then B".Hájek and Hall (1994) give a historical summary. The debate was actually framed as being about the probabilities of conditional sentences, rather than conditional events.
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Subcontraries, which medieval logicians represented in the form 'quoddam A est B' (some particular A is B) and 'quoddam A non est B' (some particular A is not B) cannot both be false, since their universal contradictory statements (every A is B / no A is B) cannot both be true. This leads to a difficulty that was first identified by Peter Abelard. 'Some A is B' seems to imply 'something is A'. For example, 'Some man is white' seems to imply that at least one thing is a man, namely the man who has to be white, if 'some man is white' is true. But, 'some man is not white' also implies that something is a man, namely the man who is not white, if the statement 'some man is not white' is true.
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The term itself came from an ancient Greek word that means "a turning back on one" and likened to the image of a snake devouring its own tail. Socrates' peritrope described Protagoras' view as a theory that no longer requires criticism because it already devours itself. Sextus Empiricus is thought to have given the name in a comment on Protagoras' view in Against the Logicians(389-90) written around 200 CE. The name has been in continuous use ever since, as Socrates' argument provides the foundation for classical propositional logic and hence much of traditional western philosophy (or analytic philosophy). For instance, Sextus noted that Protagoras' famous doctrine that man is the measure of all things is self- refuting because "one of the things that appears (is judged) to be the case is that not every appearance is true".
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In opposition to Alfred Tarski and other anti-psychologist logicians who taught that the natural language leads to contradiction by its very nature, he rejected this conviction and proposed a formal system of the Universal Syntax which imitates the versatility of colloquial language. On his approach, the axiomatization of quotation-operator is the best device which allows to marry logic with metalogic and prove the adequacy theorem for the notion of truth. He dealt with the problem on invasion of logic by postmodernity, because the deconstruction of logical rules is not at stake, there would be no logic, while the foundation of logic could be rocked. Despite that in the mathematical genealogy he was and a son of Andrzej Mostowski and a grandson of Alfred Tarski, he manifestly undermined and rebutted the classical anti-psychologism of modern logic.
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In the area of education and learning, one of Hakim's most important contributions was the founding in 1005 of the Dar al-Alem (House of Knowledge) or Dar al-Hikma (House of Wisdom).Maqrizi, 1853–54, 1995; Halm, 1997, pp. 71–78 A wide range of subjects ranging from the Qur'an and hadith to philosophy and astronomy were taught at the Dar al-alem, which was equipped with a vast library. Access to education was made available to the public and many Fatimid da'is received at least part of their training in this major institution of learning which served the Ismaili da'wa (mission) until the downfall of the Fatimid dynasty. For more than 100 years, Dar al-‘Ilm distinguished itself as a center of learning where astronomers, mathematicians, grammarians, logicians, physicians, philologists, jurists and others conducted research, gave lectures and collaborated.
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After studying in Liège with Philippe Devaux, a translator and friend of Russell, and later in 1971 at Harvard University, he was appointed an ordinary professor at the University of Liège in 1972. There, he taught logic and English-language philosophy. He was a Research Fellow at Stanford and Berkeley universities from 1974 to 1975, an invited professor at the Collège de France in 1981, Research Fellow at the Research School of Social Sciences in Canberra in 1984, and holder of the Francqui Chair at the University of Ghent in 1988. He maintained a long-standing connection with the Belgian National Center for Research in Logic that was founded in 1955 by R. Feys, Ch. Perelman, A. Borgers, A. Bayart, PhDevaux, and others, which maintained close connections with E. W. Beth and other logicians in The Netherlands.
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The work of Chomsky in generative linguistics apparently inspired much more confidence in philosophers and logicians to assert that perhaps natural languages weren't as unsystematic and misleading as their philosophical predecessors had made them out to be ... at the end of 1960s formal semantics began to flourish." writes: "Recent work by Chomsky and others is doing much to bring the complexities of natural languages within the scope of serious semantic theory". ;Computer science With its formal and logical treatment of language, Syntactic Structures also brought linguistics and the new field of computer science closer together. Computer scientist Donald Knuth (winner of the Turing Award) recounted that he read Syntactic Structures in 1961 and was influenced by it.From the preface of : "... researchers in linguistics were beginning to formulate rules of grammar that were considerably more mathematical than before.
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4, p305–328 contributed to Persia emerging as what culminated into the "Islamic Golden Age". During this period, hundreds of scholars and scientists vastly contributed to technology, science and medicine, later influencing the rise of European science during the Renaissance.Kühnel E., in Zeitschrift der deutschen morgenländischen Gesell, Vol. CVI (1956) The most important scholars of almost all of the Islamic sects and schools of thought were Persian or live in Iran including most notable and reliable Hadith collectors of Shia and Sunni like Shaikh Saduq, Shaikh Kulainy, Imam Bukhari, Imam Muslim and Hakim al-Nishaburi, the greatest theologians of Shia and Sunni like Shaykh Tusi, Imam Ghazali, Imam Fakhr al- Razi and Al-Zamakhshari, the greatest physicians, astronomers, logicians, mathematicians, metaphysicians, philosophers and scientists like Al-Farabi, Avicenna, and Nasīr al-Dīn al-Tūsī, the greatest Shaykh of Sufism like Rumi, Abdul-Qadir Gilani.
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Early forms of analogical reasoning, inductive reasoning and categorical syllogism were introduced in Fiqh (Islamic jurisprudence), Sharia (Islamic law) and Kalam (Islamic theology) from the 7th century with the process of Qiyas, before the Arabic translations of Aristotle's works. Later during the Islamic Golden Age, there was a logical debate among Islamic philosophers, logicians and theologians over whether the term Qiyas refers to analogical reasoning, inductive reasoning or categorical syllogism. Some Islamic scholars argued that Qiyas refers to inductive reasoning, which Ibn Hazm (994-1064) disagreed with, arguing that Qiyas does not refer to inductive reasoning, but refers to categorical syllogism in a real sense and analogical reasoning in a metaphorical sense. On the other hand, al-Ghazali (1058–1111) (and in modern times, Abu Muhammad Asem al-Maqdisi) argued that Qiyas refers to analogical reasoning in a real sense and categorical syllogism in a metaphorical sense.
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Chinese philosophy originates in the Spring and Autumn period (春秋) and Warring States period (战国), during a period known as the "Hundred Schools of Thought", which was characterized by significant intellectual and cultural developments. Although much of Chinese philosophy begins in the Warring States period, elements of Chinese philosophy have existed for several thousand years; some can be found in the Yi Jing (the Book of Changes), an ancient compendium of divination, which dates back to at least 672 BCE.page 60, Great Thinkers of the Eastern World, edited Ian McGreal Harper Collins 1995, It was during the Warring States era that what Sima Tan termed the major philosophical schools of China—Confucianism, Legalism, and Taoism—arose, along with philosophies that later fell into obscurity, like Agriculturalism, Mohism, Chinese Naturalism, and the Logicians. Even in modern society, Confucianism is still the creed of etiquette for Chinese society.
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An extensive bibliography is included in (Wheeler 2007). Philosophical logicians and AI researchers have tended to be interested in reconciling weakened versions of the three principles, and there are many ways to do this, including Jim Hawthorne and Luc Bovens's (1999) logic of belief, Gregory Wheeler's (2006) use of 1-monotone capacities, Bryson Brown's (1999) application of preservationist para-consistent logics, Igor Douven and Timothy Williamson's (2006) appeal to cumulative non-monotonic logics, Horacio Arlo-Costa's (2007) use of minimal model (classical) modal logics, and Joe Halpern's (2003) use of first-order probability. Finally, philosophers of science, decision scientists, and statisticians are inclined to see the lottery paradox as an early example of the complications one faces in constructing principled methods for aggregating uncertain information, which is now a discipline of its own, with a dedicated journal, Information Fusion, in addition to continuous contributions to general area journals.
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Eventually logicians found that restricting Frege's logic in various ways—to what is now called first-order logic—eliminated this problem: sets and properties cannot be quantified over in first-order-logic alone. The now-standard hierarchy of orders of logics dates from this time. It was found that set theory could be formulated as an axiomatized system within the apparatus of first-order logic (at the cost of several kinds of completeness, but nothing so bad as Russell's paradox), and this was done (see Zermelo–Fraenkel set theory), as sets are vital for mathematics. Arithmetic, mereology, and a variety of other powerful logical theories could be formulated axiomatically without appeal to any more logical apparatus than first-order quantification, and this, along with Gödel and Skolem's adherence to first-order logic, led to a general decline in work in second (or any higher) order logic.
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Early forms of analogical reasoning, inductive reasoning and categorical syllogism were introduced in Fiqh (Islamic jurisprudence), Sharia (Islamic law) and Kalam (Islamic theology) from the 7th century with the process of Qiyas, before the Arabic translations of Aristotle's works. Later during the Islamic Golden Age, there was a logical debate among Islamic philosophers, logicians and theologians over whether the term Qiyas refers to analogical reasoning, inductive reasoning or categorical syllogism. Some Islamic scholars argued that Qiyas refers to inductive reasoning, which Ibn Hazm (994-1064) disagreed with, arguing that Qiyas does not refer to inductive reasoning, but refers to categorical syllogism in a real sense and analogical reasoning in a metaphorical sense. On the other hand, al-Ghazali (1058–1111) (and in modern times, Abu Muhammad Asem al-Maqdisi) argued that Qiyas refers to analogical reasoning in a real sense and categorical syllogism in a metaphorical sense.
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He praised Nagel for the thoroughness of his treatment of the nature of scientific inquiry, his discussion of explanation in the biological sciences, his criticism of functionalism in the social sciences, and his discussion of historical explanation. Scriven described the book as a "great work", and considered Nagel's treatment of some subjects definitive. He praised Nagel's discussion of the history of science and careful analysis of "alternative positions", pointing in particular to Nagel's "discussion of the ontological status of theories and models" and "his treatment of fallacious arguments for holism"; he also complimented Nagel for his criticism of Berlin and his discussion of the meaning of scientific laws. However, he noted that the book was not easy to read; he also criticized Nagel for being too willing to accept the analyses of certain concepts proposed by symbolic logicians, for failing to fully pursue the implications of his ideas about scientific practice, giving his treatment of historical explanation as an example.
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Sophismata (plural form of the Greek word σόφισμα, 'sophisma', which also gave rise to the related term "sophism") in medieval philosophy are difficult or puzzling sentences presenting difficulties of logical analysis that must be solved. Sophismata-literature grew in importance during the thirteenth and fourteenth centuries, and many important developments in philosophy (particularly in logic and natural philosophy) occurred as a result of investigation into their logical and semantic properties. Sophismata are "ambiguous, puzzling or simply difficult sentences" that were used by Medieval logicians for educational purposes and for disputation about logic. Sophismata were written in Latin and the meaning of many of them is lost when translated into other languages. They can be divided into sentences that: #are odd or have odd consequences #are ambiguous, and can be true or false according to the interpretation we give it, or #have nothing special about them in itself, but become puzzling when they occur in definite contexts (or “cases”, casus).
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Set theory began with the work of the logicians with the notion of "class" (modern "set") for example , Jevons (1880), , and . It was given a push by Georg Cantor's attempt to define the infinite in set-theoretic treatment (1870–1890) and a subsequent discovery of an antinomy (contradiction, paradox) in this treatment (Cantor's paradox), by Russell's discovery (1902) of an antinomy in Frege's 1879 (Russell's paradox), by the discovery of more antinomies in the early 20th century (e.g., the 1897 Burali- Forti paradox and the 1905 Richard paradox), and by resistance to Russell's complex treatment of logic and dislike of his axiom of reducibility"The nonprimitive and arbitrary character of this axiom drew forth severe criticism, and much of subsequent refinement of the logistic program lies in attempts to devise some method of avoiding the disliked axiom of reducibility" . (1908, 1910–1913) that he proposed as a means to evade the antinomies.
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The expressions mentioned above all have been used in many other ways. Many other propositions have also been mentioned as laws of thought, including the dictum de omni et nullo attributed to Aristotle, the substitutivity of identicals (or equals) attributed to Euclid, the so-called identity of indiscernibles attributed to Gottfried Wilhelm Leibniz, and other "logical truths". The expression "laws of thought" gained added prominence through its use by Boole (1815–64) to denote theorems of his "algebra of logic"; in fact, he named his second logic book An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities (1854). Modern logicians, in almost unanimous disagreement with Boole, take this expression to be a misnomer; none of the above propositions classed under "laws of thought" are explicitly about thought per se, a mental phenomenon studied by psychology, nor do they involve explicit reference to a thinker or knower as would be the case in pragmatics or in epistemology.
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Despite the logical sophistication of al-Ghazali, the rise of the Ash'ari school in the 12th century slowly suffocated original work on logic in much of the Islamic world, though logic continued to be studied in some Islamic regions such as Persia and the Levant. Fakhr al-Din al-Razi (b. 1149) criticised Aristotle's "first figure" and developed a form of inductive logic, foreshadowing the system of inductive logic developed by John Stuart Mill (1806–1873). Systematic refutations of Greek logic were written by the Illuminationist school, founded by Shahab al-Din Suhrawardi (1155–1191), who developed the idea of "decisive necessity", an important innovation in the history of logical philosophical speculation,Another systematic refutation of Greek logic was written by Ibn Taymiyyah (1263–1328), the Ar-Radd 'ala al-Mantiqiyyin (Refutation of Greek Logicians), where he argued against the usefulness, though not the validity, of the syllogism See pp. 253–54 of and in favour of inductive reasoning.
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A common element of such axiomatizations is the assumption, shared with inclusion, that the part- whole relation orders its universe, meaning that everything is a part of itself (reflexivity), that a part of a part of a whole is itself a part of that whole (transitivity), and that two distinct entities cannot each be a part of the other (antisymmetry), thus forming a poset. A variant of this axiomatization denies that anything is ever part of itself (irreflexivity) while accepting transitivity, from which antisymmetry follows automatically. Although mereology is an application of mathematical logic, what could be argued to be a sort of "proto-geometry", it has been wholly developed by logicians, ontologists, linguists, engineers, and computer scientists, especially those working in artificial intelligence. In particular, mereology is also on the basis for a point-free foundation of geometry (see for example the quoted pioneering paper of Alfred Tarski and the review paper by Gerla 1995).
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Despite the logical sophistication of al-Ghazali, the rise of the Ash'ari school in the 12th century slowly suffocated original work on logic in much of the Islamic world, though logic continued to be studied in some Islamic regions such as Persia and the Levant. Fakhr al-Din al-Razi (b. 1149) criticised Aristotle's "first figure" and developed a form of inductive logic, foreshadowing the system of inductive logic developed by John Stuart Mill (1806–1873). Systematic refutations of Greek logic were written by the Illuminationist school, founded by Shahab al-Din Suhrawardi (1155–1191), who developed the idea of "decisive necessity", an important innovation in the history of logical philosophical speculation. Another systematic refutation of Greek logic was written by Ibn Taymiyyah (1263-1328), the Ar-Radd 'ala al-Mantiqiyyin (Refutation of Greek Logicians), where he argued against the usefulness, though not the validity, of the syllogismSee pp. 253-254 of and in favour of inductive reasoning.
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Roughly, a reduction strategy is a function that maps a lambda calculus term with reducible expressions to one particular reducible expression, the one to be reduced next. Mathematical logicians have studied the properties of this system for decades, and the superficial similarity between the description of evaluation strategies and reduction strategies originally misled programming language researchers to speculate that the two were identical, a belief that is still visible in popular textbooks from the early 1980s;Structure and Interpretation of Computer Programs by Abelson and Sussman, MIT Press 1983 these are, however, different concepts. PlotkinCall- by-name, call-by-value, and the lambda calculus showed in 1973, however, that a proper model of an evaluation strategy calls for the formulation of a new axiom for function calls, that is, an entirely new calculus. He validates this idea with two different calculi: one for call-by-name and another one for call-by-value, each for purely functional programming languages.
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Boole was apparently disconcerted at the book's reception just as a mathematical toolset: > George afterwards learned, to his great joy, that the same conception of the > basis of Logic was held by Leibniz, the contemporary of Newton. De Morgan, > of course, understood the formula in its true sense; he was Boole's > collaborator all along. Herbert Spencer, Jowett, and Robert Leslie Ellis > understood, I feel sure; and a few others, but nearly all the logicians and > mathematicians ignored [953] the statement that the book was meant to throw > light on the nature of the human mind; and treated the formula entirely as a > wonderful new method of reducing to logical order masses of evidence about > external fact. Mary Boole claimed that there was profound influence – via her uncle George Everest – of Indian thought in general and Indian logic, in particular, on George Boole, as well as on Augustus De Morgan and Charles Babbage:Kak, S. (2018) George Boole’s Laws of Thought and Indian logic.
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Hintikka's proposal was met with skepticism by a number of logicians because some first-order sentences like the one below appear to capture well enough the natural language Hintikka sentence. : [\forall x_1 \, \exists y_1 \, \forall x_2 \, \exists y_2\, \varphi (x_1, x_2, y_1, y_2)] \wedge [\forall x_2 \, \exists y_2 \, \forall x_1 \, \exists y_1\, \varphi (x_1, x_2, y_1, y_2)] where : \varphi (x_1, x_2, y_1, y_2) denotes : (V(x_1) \wedge T(x_2)) \rightarrow (R(x_1,y_1) \wedge R(x_2,y_2) \wedge H(y_1, y_2) \wedge H(y_2, y_1)) Although much purely theoretical debate followed, it wasn't until 2009 that some empirical tests with students trained in logic found that they are more likely to assign models matching the "bidirectional" first-order sentence rather than branching-quantifier sentence to several natural-language constructs derived from the Hintikka sentence. For instance students were shown undirected bipartite graphs--with squares and circles as vertices--and asked to say whether sentences like "more than 3 circles and more than 3 squares are connected by lines" were correctly describing the diagrams.
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Logicians of this time were primarily involved with analyzing syllogisms (the 2000-year-old Aristotelian forms and otherwise), or as Augustus De Morgan (1847) stated it: "the examination of that part of reasoning which depends upon the manner in which inferences are formed, and the investigation of general maxims and rules for constructing arguments". At this time the notion of (logical) "function" is not explicit, but at least in the work of De Morgan and George Boole it is implied: we see abstraction of the argument forms, the introduction of variables, the introduction of a symbolic algebra with respect to these variables, and some of the notions of set theory. De Morgan's 1847 "FORMAL LOGIC OR, The Calculus of Inference, Necessary and Probable" observes that "[a] logical truth depends upon the structure of the statement, and not upon the particular matters spoken of"; he wastes no time (preface page i) abstracting: "In the form of the proposition, the copula is made as abstract as the terms". He immediately (p.
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Metaphysics 7, 1011b 26–27 Aristotle wrote that ambiguity can arise from the use of ambiguous names, but cannot exist in the facts themselves: Aristotle's assertion that "...it will not be possible to be and not to be the same thing", which would be written in propositional logic as ¬(P ∧ ¬P), is a statement modern logicians could call the law of excluded middle (P ∨ ¬P), as distribution of the negation of Aristotle's assertion makes them equivalent, regardless that the former claims that no statement is both true and false, while the latter requires that any statement is either true or false. However, Aristotle also writes, "since it is impossible that contradictories should be at the same time true of the same thing, obviously contraries also cannot belong at the same time to the same thing" (Book IV, CH 6, p. 531). He then proposes that "there cannot be an intermediate between contradictories, but of one subject we must either affirm or deny any one predicate" (Book IV, CH 7, p. 531).
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Anvaya refers to the logical connection of words, as to how different words relate with each other to convey a significant meaning or idea. Literally, Anvaya (Sanskrit: अन्वय) means - positive; affirmative or nexus; but in grammar and logic this word refers to - 'concordance' or 'agreement', such as the agreement which exists between two things that are present, as between 'smoke' and 'fire', it is universally known that - "where there is smoke, there is fire". However, this word is commonly used in Sanskrit grammar and logic along with the word, Vyatireka, which means - agreement in absence between two things, such as absence of 'smoke' and 'fire' - "where there is no smoke, there is no fire". Anvaya-vyatireka, is the term used by the Buddhists and Hindu logicians as a dual procedure - to signify 'separation' and 'connection', and to indicate a type of inference in which hetu (reason) is co-present or is co-absent with sādhya (major term), as the pair of positive and negative instantiations which represent the inductive and the deductive reasoning, both.
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