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True Contradiction
Dialetheism (from Greek 'twice' and 'truth') is the view that there are statements that are both true and false. More precisely, it is the belief that there can be a true statement whose negation is also true. Such statements are called "true contradictions", ''dialetheia'', or nondualisms. Dialetheism is not a system of formal logic; instead, it is a thesis about truth that influences the construction of a formal logic, often based on pre-existing systems. Introducing dialetheism has various consequences, depending on the theory into which it is introduced. A common mistake resulting from this is to reject dialetheism on the basis that, in traditional systems of logic (e.g., classical logic and intuitionistic logic), every statement becomes a theorem if a contradiction is true, trivialising such systems when dialetheism is included as an axiom.Ben Burgis, Visiting Professor of Philosophy at the University of Ulsan in South Korea, iBlog&~Blog Other logical systems, however ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic period (), and the Classical period (). Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers. It has contributed many words to English vocabulary and has been a standard subject of study in educational institutions of the Western world since the Renaissance. This article primarily contains information about the Epic and Classical periods of the language. From the Hellenistic period (), Ancient Greek was followed by Koine Greek, which is regarded as a separate historical stage, although its earliest form closely resembles Attic Greek and its latest form approaches Medieval Greek. There were several regional dialects of Ancient Greek, of which Attic Greek developed into Koine. Dia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russell's Paradox
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, which he told Hilbert and Richard Dedekind by letter. According to the unrestricted comprehension principle, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. Let ''R'' be the set of all sets that are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: 2^=\aleph_1, or even shorter with beth numbers: \beth_1 = \aleph_1. The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940. The name of the hypothesis comes from the term '' the continuum'' for the real numbers. History Cantor believed the continuum hypothesis to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-ordering Theorem
In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set ''X'' is ''well-ordered'' by a strict total order if every non-empty subset of ''X'' has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also ). Ernst Zermelo introduced the axiom of choice as an "unobjectionable logical principle" to prove the well-ordering theorem. One can conclude from the well-ordering theorem that every set is susceptible to transfinite induction, which is considered by mathematicians to be a powerful technique. One famous consequence of the theorem is the Banach–Tarski paradox. History Georg Cantor considered the well-ordering theorem to be a "fundamental principle of thought". However, it is considered difficult or even impossible to visualize a well-ordering of \mathbb; such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logicism
In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano. Overview Dedekind's path to logicism had a turning point when he was able to construct a model satisfying the axioms characterizing the real numbers using certain sets of rational numbers. This and related ideas convinced him that arithmetic, algebra and analysis were reducible to the natural numbers plus a "logic" of classes. Furthermore by 1872 he had concluded that the naturals themselves were reducible to sets and mappings. It is likely that other logicists, most importantly Frege, were also guided by the new theories of the real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Logic Symbols
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics. Additionally, the subsequent columns contains an informal explanation, a short example, the Unicode location, the name for use in HTML documents, and the LaTeX symbol. Basic logic symbols Advanced and rarely used logical symbols These symbols are sorted by their Unicode value: Usage in various countries Poland and Germany in Poland, the universal quantifier is sometimes written ∧, and the existential quantifier as ∨. The same applies for Germany. Japan The ⇒ symbol is often used in text to mean "result" or "conclusion", as in "We examined whether to sell the product ⇒ We will not sell it". Also, the → symbol is often used to denote "changed to", as in the sentence "The interest rate changed. March 20% → April 21%". See also * Józef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bimal Krishna Matilal
Bimal Krishna Matilal (1 June 1935 – 8 June 1991) was an eminent British-Indian philosopher whose writings presented the Indian philosophical tradition as a comprehensive system of logic incorporating most issues addressed by themes in Western philosophy. From 1977 to 1991, he was the Spalding Professor of Eastern Religion and Ethics at the University of Oxford. Education Literate in Sanskrit from an early age, Matilal was also drawn towards Mathematics and Logic. He was trained in the traditional Indian philosophical system by leading scholars of the Sanskrit College, where he himself was a teacher from 1957 to 1962. He was taught by scholars like pandit Taranath Tarkatirtha and Kalipada Tarkacharya. He also interacted with pandit Ananta Kumar Nyayatarkatirtha, Madhusudan Nyayacharya and Visvabandhu Tarkatirtha. The ''upadhi'' (degree) of Tarkatirtha (master of Logic) was awarded to him in 1962. While teaching at the Sanskrit College (an affiliated college of the University ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catuṣkoṭi
''Catuṣkoṭi'' (Sanskrit; Devanagari: चतुष्कोटि, , Sinhalese:චතුස්කෝටිකය) is a logical argument(s) of a 'suite of four discrete functions' or 'an indivisible quaternity' that has multiple applications and has been important in the Dharmic traditions of Indian logic, the Buddhist logico-epistemological traditions, particularly those of the Madhyamaka school, and in the skeptical Greek philosophy of Pyrrhonism. In particular, the catuṣkoṭi is a "four-cornered" system of argumentation that involves the systematic examination of each of the 4 possibilities of a proposition, ''P'': # ''P''; that is being. # not ''P''; that is not being. # ''P'' and not ''P''; that is being and that is not being. # not (''P'' or not ''P''); that is neither not being nor is that being. These four statements hold the following properties: (1) each alternative is mutually exclusive (that is, one of, but no more than one of, the four statements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buddhist
Buddhism ( , ), also known as Buddha Dharma and Dharmavinaya (), is an Indian religion or philosophical tradition based on teachings attributed to the Buddha. It originated in northern India as a -movement in the 5th century BCE, and gradually spread throughout much of Asia via the Silk Road. It is the world's fourth-largest religion, with over 520 million followers (Buddhists) who comprise seven percent of the global population. The Buddha taught the Middle Way, a path of spiritual development that avoids both extreme asceticism and hedonism. It aims at liberation from clinging and craving to things which are impermanent (), incapable of satisfying ('), and without a lasting essence (), ending the cycle of death and rebirth (). A summary of this path is expressed in the Noble Eightfold Path, a training of the mind with observance of Buddhist ethics and meditation. Other widely observed practices include: monasticism; " taking refuge" in the Buddha, the , and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anekantavada
( hi, अनेकान्तवाद, "many-sidedness") is the Jain doctrine about metaphysical truths that emerged in ancient India. It states that the ultimate truth and reality is complex and has multiple aspects. According to Jainism, no single, specific statement can describe the nature of existence and the absolute truth. This knowledge ('' Kevala Jnana''), it adds, is comprehended only by the Arihants. Other beings and their statements about absolute truth are incomplete, and at best a partial truth. All knowledge claims, according to the ''anekāntavāda'' doctrine must be qualified in many ways, including being affirmed and denied. Anekāntavāda is a fundamental doctrine of Jainism. The origins of ''anekāntavāda '' can be traced back to the teachings of Mahāvīra (599–527 BCE), the 24th Jain . The dialectical concepts of ''syādvāda'' "conditioned viewpoints" and ''nayavāda'' "partial viewpoints" arose from ''anekāntavāda'' in the medieval era, providin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jain
Jainism ( ), also known as Jain Dharma, is an Indian religion. Jainism traces its spiritual ideas and history through the succession of twenty-four tirthankaras (supreme preachers of ''Dharma''), with the first in the current time cycle being Rishabhadeva, whom the tradition holds to have lived millions of years ago, the twenty-third ''tirthankara'' Parshvanatha, whom historians date to the 9th century BCE, and the twenty-fourth ''tirthankara'' Mahavira, around 600 BCE. Jainism is considered to be an eternal ''dharma'' with the ''tirthankaras'' guiding every time cycle of the cosmology. The three main pillars of Jainism are ''ahiṃsā'' (non-violence), ''anekāntavāda'' (non-absolutism), and '' aparigraha'' (asceticism). Jain monks, after positioning themselves in the sublime state of soul consciousness, take five main vows: ''ahiṃsā'' (non-violence), '' satya'' (truth), '' asteya'' (not stealing), ''brahmacharya'' (chastity), and '' aparigraha'' (non-possessiveness). Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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