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Boolean Domain
In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include ''false'' and ''true''. In logic, mathematics and theoretical computer science, a Boolean domain is usually written as , or \mathbb. The algebraic structure that naturally builds on a Boolean domain is the Boolean algebra with two elements. The initial object in the category of bounded lattices is a Boolean domain. In computer science, a Boolean variable is a variable that takes values in some Boolean domain. Some programming languages feature reserved words or symbols for the elements of the Boolean domain, for example false and true. However, many programming languages do not have a Boolean datatype in the strict sense. In C or BASIC, for example, falsity is represented by the number 0 and truth is represented by the number 1 or −1, and all variables that can take these values can also take any other numerical values. Generalizations The Boo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: , , and . However, the notation ' is most commonly reserved for the closed interval . Properties The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space, it is compact, contractible, path connected and locally path connected. The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval. In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eric C
The given name Eric, Erich, Erikk, Erik, Erick, or Eirik is derived from the Old Norse name ''Eiríkr'' (or ''Eríkr'' in Old East Norse due to monophthongization). The first element, ''ei-'' may be derived from the older Proto-Norse ''* aina(z)'', meaning "one, alone, unique", ''as in the form'' ''Æ∆inrikr'' explicitly, but it could also be from ''* aiwa(z)'' "everlasting, eternity", as in the Gothic form ''Euric''. The second element ''- ríkr'' stems either from Proto-Germanic ''* ríks'' "king, ruler" (cf. Gothic ''reiks'') or the therefrom derived ''* ríkijaz'' "kingly, powerful, rich, prince"; from the common Proto-Indo-European root * h₃rḗǵs. The name is thus usually taken to mean "sole ruler, autocrat" or "eternal ruler, ever powerful". ''Eric'' used in the sense of a proper noun meaning "one ruler" may be the origin of ''Eriksgata'', and if so it would have meant "one ruler's journey". The tour was the medieval Swedish king's journey, when newly elected, to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliott Mendelson
Elliott Mendelson (May 24, 1931 – May 7, 2020) was an American logician. He was a professor of mathematics at Queens College of the City University of New York, and the Graduate Center, CUNY. He was Jr. Fellow, Society of Fellows, Harvard University, 1956–58. Career Mendelson earned his BA from Columbia University and PhD from Cornell University. Mendelson taught mathematics at the college level for more than 30 years, and is the author of books on logic, philosophy of mathematics, calculus, game theory and mathematical analysis. His ''Introduction to Mathematical Logic'', first published in 1964, was reviewed by Dirk van Dalen who noted that it included "a large variety of subjects that should be part of the education of any mathematics student with an interest in foundational matters." Dirk van Dalen (1969Review: Introduction to Mathematical Logic Journal of Symbolic Logic 34(1): 110,1 via JSTOR Books Sole author * * * * * * * Co-author * * * * Editor * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard C
Richard is a male given name. It originates, via Old French, from Old Frankish and is a compound of the words descending from Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include " Richie", " Dick", " Dickon", " Dickie", "Rich", " Rick", "Rico", " Ricky", and more. Richard is a common English, German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Catalan "Ricard" and the Italian "Riccardo", among others (see comprehensive variant list below). People named Richard Multiple people with the same name * Richard Andersen (other) * Richard Anderson (other) * Richard Cartwright (disambiguati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Boolos
George Stephen Boolos (; 4 September 1940 – 27 May 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology. Life Boolos is of Greek-Jewish descent. He graduated with an A.B. in mathematics from Princeton University after completing a senior thesis, titled "A simple proof of Gödel's first incompleteness theorem", under the supervision of Raymond Smullyan. Oxford University awarded him the B.Phil. in 1963. In 1966, he obtained the first PhD in philosophy ever awarded by the Massachusetts Institute of Technology, under the direction of Hilary Putnam. After teaching three years at Columbia University, he returned to MIT in 1969, where he spent the rest of his career. A charismatic speaker well known for his clarity and wit, he once delivered a lecture (1994b) giving an account of Gödel's second incompleteness theorem, employing only words of one syllable. At the end of his viva, Hilary Putnam asked him, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Makinson
David Clement Makinson (born 27 August 1941), is an Australian mathematical logician living in London, England. Career Makinson began his studies at Sydney University in 1958 and was an associate of the Libertarian Society and Sydney Push. He is a Visiting Professor in the London School of Economics, University of London, and an associate member of the Centre de Recherche en Epistémologie Appliquée (CREA), École Polytechnique, Paris. He has held professorial rank positions in King's College London, University of London and in the American University of Beirut, Lebanon. From 1980 till 2001 he worked for UNESCO, Paris. Contributions David Makinson is highly regarded for his work on belief revision, uncertain reasoning, and modal logic. While studying in Oxford University (Worcester College) for his D.Phil under the supervision of Michael Dummett, he identified the preface paradox. In belief revision he created the AGM account of theory change with Carlos Alchour ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirk Van Dalen
Dirk van Dalen (born 20 December 1932, Amsterdam) is a Dutch mathematician and historian of science. Van Dalen studied mathematics and physics and astronomy at the University of Amsterdam. Inspired by the work of Brouwer and Heyting, he received his Ph.D. in 1963 from the University of Amsterdam for the thesis ''Extension problems in intuitionistic plane Projective geometry.'' From 1964 to 1966 Van Dalen taught logic and mathematics at MIT, and later Oxford. From 1967 he was professor at the University of Utrecht. In 2003 Dirk van Dalen was awarded the Academy Medal 2003 of the Royal Dutch Academy of Sciences for bringing the works of Brouwer to international attention. Works * 1958: (with Yehoshua Bar-Hillel and Azriel Levy) ''Foundations of Set Theory'', North Holland Publishing * 1963: Extension problems in intuitionistic plane projective geometry * 1978: (with H.C. Doets and H. De Swart) ''Sets: Naive, Axiomatic and Applied'', Pergamon Press * 1980: ''Logic and Structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GF(2)
(also denoted \mathbb F_2, or \mathbb Z/2\mathbb Z) is the finite field of two elements (GF is the initialism of ''Galois field'', another name for finite fields). Notations and \mathbb Z_2 may be encountered although they can be confused with the notation of -adic integers. is the field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity are denoted respectively and , as usual. The elements of may be identified with the two possible values of a bit and to the boolean values ''true'' and ''false''. It follows that is fundamental and ubiquitous in computer science and its logical foundations. Definition GF(2) is the unique field with two elements with its additive and multiplicative identities respectively denoted and . Its addition is defined as the usual addition of integers but modulo 2 and corresponds to the table below: If the elements of GF(2) are seen as boolean values, then the addition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean-valued Function
A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f : X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set, (for example B = ), whose elements are interpreted as logical values, for example, 0 = false and 1 = true, i.e., a single bit of information. In the formal sciences, mathematics, mathematical logic, statistics, and their applied disciplines, a Boolean-valued function may also be referred to as a characteristic function, indicator function, predicate, or proposition. In all of these uses, it is understood that the various terms refer to a mathematical object and not the corresponding semiotic sign or syntactic expression. In formal semantic theories of truth, a truth predicate is a predicate on the sentences of a formal language, interpreted for logic, that formalizes the intuitive concept that is normally expressed by saying that a sentence is true. A truth predicate may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probabilistic Logic
Probabilistic logic (also probability logic and probabilistic reasoning) involves the use of probability and logic to deal with uncertain situations. Probabilistic logic extends traditional logic truth tables with probabilistic expressions. A difficulty of probabilistic logics is their tendency to multiply the computational complexities of their probabilistic and logical components. Other difficulties include the possibility of counter-intuitive results, such as in case of belief fusion in Dempster–Shafer theory. Source trust and epistemic uncertainty about the probabilities they provide, such as defined in subjective logic, are additional elements to consider. The need to deal with a broad variety of contexts and issues has led to many different proposals. Logical background There are numerous proposals for probabilistic logics. Very roughly, they can be categorized into two different classes: those logics that attempt to make a probabilistic extension to logical entailment, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or 1. The term ''fuzzy logic'' was introduced with the 1965 proposal of fuzzy set theory by Iranian Azerbaijani mathematician Lotfi Zadeh. Fuzzy logic had, however, been studied since the 1920s, as infinite-valued logic—notably by Łukasiewicz and Tarski. Fuzzy logic is based on the observation that people make decisions based on imprecise and non-numerical information. Fuzzy models or sets are mathematical means of representing vagueness and imprecise information (hence the term fuzzy). These models have the capability of recognising, representing, manipulating, interpreting, and using data and information that are vague and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
