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Extensional Equality
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal definitions of objects are the same. In mathematics The extensional definition of function equality, discussed above, is commonly used in mathematics. A similar extensional definition is usually employed for relations: two relations are said to be equal if they have the same extensions. In set theory, the axiom of extensionality states that two sets are equal if and only if they contain the same elements. In mathematics formalized in set theory, it is common to identify relations—and, most importantly, functions—with their extension as stated above, so that it is impossible for two relations or functions with the same extension to be distinguished. Other mathematical objects are also constructed in such a way that the intui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decidability (logic)
In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not. Logical systems are decidable if membership in their set of logically valid formulas (or theorems) can be effectively determined. A theory (set of sentences closed under logical consequence) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems are undecidable, that is, it has been proven that no effective method for determining membership (returning a correct answer after finite, though possibly very long, time in all cases) can exist for them. Decidability of a logical system Each logical system comes with both a syntactic component, which among other things determines the notion of provability, and a semantic component, which determine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of ''naive set theory''. After the discovery of Paradoxes of set theory, paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NLab
The ''n''Lab is a wiki for research-level notes, expositions and collaborative work, including original research, in mathematics, physics, and philosophy, with a focus on methods from type theory, category theory, and homotopy theory. The ''n''Lab espouses the "''n''-point of view" (a deliberate pun on Wikipedia's "neutral point of view") that type theory, homotopy theory, category theory, and higher category theory provide a useful unifying viewpoint for mathematics, physics and philosophy. The ''n'' in ''n''-point of view could refer to either ''n''-categories as found in higher category theory, ''n''-groupoids as found in both homotopy theory and higher category theory, or ''n''-types as found in homotopy type theory. Overview The ''n''Lab was originally conceived to provide a repository for ideas (and even new research) generated in the comments on posts at the ''n''-Category Café, a group blog run (at the time) by John C. Baez, David Corfield and Urs Schreiber. Eventua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equality
Equality generally refers to the fact of being equal, of having the same value. In specific contexts, equality may refer to: Society * Egalitarianism, a trend of thought that favors equality for all people ** Political egalitarianism, in which all members of a society are of equal standing * Equal opportunity, a stipulation that all people should be treated similarly * Equality of outcome, in which the general conditions of people's lives are similar * Substantive equality, Equality of outcome for groups * For specific groups: ** Gender equality ** Racial equality * Social equality, in which all people within a group have the same status * Economic inequality * Equality Party (other), several political parties * Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elites Law * Equality before the law, the principle under which all people are subject to the same laws * Equality Act (disambi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harvard University Press
Harvard University Press (HUP) is an academic publishing house established on January 13, 1913, as a division of Harvard University. It is a member of the Association of University Presses. Its director since 2017 is George Andreou. The press maintains offices in Cambridge, Massachusetts, near Harvard Square, and in London, England. The press co-founded the distributor TriLiteral LLC with MIT Press and Yale University Press. TriLiteral was sold to LSC Communications in 2018. Notable authors published by HUP include Eudora Welty, Walter Benjamin, E. O. Wilson, John Rawls, Emily Dickinson, Stephen Jay Gould, Helen Vendler, Carol Gilligan, Amartya Sen, David Blight, Martha Nussbaum, and Thomas Piketty. The Display Room in Harvard Square, dedicated to selling HUP publications, closed on June 17, 2009. Related publishers, imprints, and series HUP owns the Belknap Press imprint (trade name), imprint, which it inaugurated in May 1954 with the publication of the ''Harvard Guide to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Synthese
''Synthese'' () is a monthly peer-reviewed academic journal covering the epistemology, methodology, and philosophy of science, and related issues. The name ''Synthese'' (from the Dutch for '' synthesis'') finds its origin in the intentions of its founding editors: making explicit the supposed internal coherence between the different, highly specialised scientific disciplines. Jaakko Hintikka was editor-in-chief from 1965 to 2002. The current editors-in-chief are Otávio Bueno (University of Miami), Wiebe van der Hoek (University of Liverpool), and Kristie Miller (University of Sydney). Editorial decision controversies In 2011, the journal became involved in a controversy over intelligent design. The printed version of the special issue ''Evolution and Its Rivals'', which appeared two years after the online version, was supplied with a disclaimer from the then editors of the journal that "appeared to undermine he authorsand the guest editors". The journal engendered controversy a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mind (journal)
''Mind'' (stylized as ''MIND'') is a quarterly peer-reviewed academic journal published by Oxford University Press on behalf of the Mind Association. Having previously published exclusively philosophy in the analytic tradition, it now "aims to take quality to be the sole criterion of publication, with no area of philosophy, no style of philosophy, and no school of philosophy excluded." Its institutional home is shared between the University of Oxford and University College London. It is considered an important resource for studying philosophy. History and profile The journal was established in 1876 by the Scottish philosopher Alexander Bain (University of Aberdeen) with his colleague and former student George Croom Robertson (University College London) as editor-in-chief. With the death of Robertson in 1891, George Stout took over the editorship and began a 'New Series'. Early on, the journal was dedicated to the question of whether psychology could be a legitimate natural s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Univalence Axiom
In mathematical logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies. This includes, among other lines of work, the construction of homotopical and higher-categorical models for such type theories; the use of type theory as a logic (or internal language) for abstract homotopy theory and higher category theory; the development of mathematics within a type-theoretic foundation (including both previously existing mathematics and new mathematics that homotopical types make possible); and the formalization of each of these in computer proof assistants. There is a large overlap between the work referred to as homotopy type theory, and that called the univalent foundations project. Although neither is precisely delineated, and the terms are sometimes used interchangeably, the choice of usage also som ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Of Indiscernibles
The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities ''x'' and ''y'' are identical if every predicate possessed by ''x'' is also possessed by ''y'' and vice versa. It states that no two distinct things (such as snowflakes) can be exactly alike, but this is intended as a metaphysical principle rather than one of natural science. A related principle is the indiscernibility of identicals, discussed below. A form of the principle is attributed to the German philosopher Gottfried Wilhelm Leibniz. While some think that Leibniz's version of the principle is meant to be only the indiscernibility of identicals, others have interpreted it as the conjunction of the identity of indiscernibles and the indiscernibility of identicals (the converse principle). Because of its association with Leibniz, the indiscernibility of identicals is sometimes known as Leibniz's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Univalent Foundations
Univalent foundations are an approach to the foundations of mathematics in which mathematical Structuralism (philosophy of mathematics), structures are built out of objects called ''types''. Types in univalent foundations do not correspond exactly to anything in set-theoretic foundations, but they may be thought of as spaces, with equal types corresponding to homotopy equivalent spaces and with equal elements of a type corresponding to points of a space connected by a path. Univalent foundations are inspired both by the old Philosophy of mathematics#Platonism, Platonic ideas of Hermann Grassmann and Georg Cantor and by "category theory, categorical" mathematics in the style of Alexander Grothendieck. Univalent foundations depart from (although are also compatible with) the use of classical predicate logic as the underlying formal Deductive reasoning, deduction system, replacing it, at the moment, with a version of Martin-Löf type theory. The development of univalent foundations is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
