|
Algebraic Logic
In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with Free variables and bound variables, free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic description of model theory, models appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics (mathematical logic), algebraic semantics for these deductive systems) and connected problems like Representation (mathematics), representation and duality. Well known results like the representation theorem for Boolean algebras and Stone duality fall under the umbrella of classical algebraic logic . Works in the more recent abstract algebraic logic (AAL) focus on the process of algebraization itself, like classifying various forms of algebraizability using the Leibniz operator . Calculus of relations A homogeneous binary relation is found in the power set of for some set ''X'', while a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebraic Logic
In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum–Tarski algebra, and how the resulting algebras are related to logical systems.Font, 2003. History The archetypal association of this kind, one fundamental to the historical origins of algebraic logic and lying at the heart of all subsequently developed subtheories, is the association between the class of Boolean algebra (structure), Boolean algebras and classical propositional calculus. This association was discovered by George Boole in the 1850s, and then further developed and refined by others, especially Charles Sanders Peirce, C. S. Peirce and Ernst Schröder (mathematician), Ernst Schröder, from the 1870s to the 1890s. This work culminated in Lindenbaum–Tarski algebras, devised by Alfred Tarski and his student Adolf Lindenbaum in the 1930s. Later, Tarski and his American students (whose ranks include Don Pigozzi) we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Matrix
A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1)-matrix is a matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets. It is an important tool in combinatorial mathematics and theoretical computer science. Matrix representation of a relation If ''R'' is a binary relation between the finite indexed sets ''X'' and ''Y'' (so ), then ''R'' can be represented by the logical matrix ''M'' whose row and column indices index the elements of ''X'' and ''Y'', respectively, such that the entries of ''M'' are defined by :m_ = \begin 1 & (x_i, y_j) \in R, \\ 0 & (x_i, y_j) \not\in R. \end In order to designate the row and column numbers of the matrix, the sets ''X'' and ''Y'' are indexed with positive integers: ''i'' ranges from 1 to the cardinality (size) of ''X'', and ''j'' ranges from 1 to the cardinality of ''Y''. See the article on indexed sets for more detail. The transpose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gunther Schmidt
Gunther Schmidt (born 1939, Rüdersdorf) is a Germans, German mathematician who works also in informatics. Life Schmidt began studying Mathematics in 1957 at Göttingen University. His academic teachers were in particular Kurt Reidemeister, Wilhelm Klingenberg and Karl Stein. In 1960 he transferred to Ludwig-Maximilians-Universität München where he studied functions of several complex variables with Karl Stein (mathematician), Karl Stein. Schmidt wrote a thesis on analytic continuation of such functions. In 1962 Schmidt began work at TU München with students of Robert Sauer, in the beginning in labs and tutorials, later in mentoring and administration. Schmidt's interests turned toward programming when he collaborated with Hans Langmaack on rewriting and the braid group in 1969. Friedrich L. Bauer and Klaus Samelson were establishing software engineering at the university and Schmidt joined their group in 1974. In 1977 he submitted his Habilitation "Programs as partial grap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Loewner
Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German. Early life and career Karl Loewner was born into a Jewish family in Lany, about 30 km from Prague, where his father Sigmund Löwner was a store owner. Loewner received his Ph.D. from the University of Prague in 1917 under supervision of Georg Pick. One of his central mathematical contributions is the proof of the Bieberbach conjecture in the first highly nontrivial case of the third coefficient. The technique he introduced, the Loewner differential equation, has had far-reaching implications in geometric function theory; it was used in the final solution of the Bieberbach conjecture by Louis de Branges in 1985. Loewner worked at the University of Berlin, University of Prague, University of Louisville, Brown University, Syracuse University and eventually at Stanford University. His students include Lipman Bers, Roger Horn, A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called the Domain of a function, domain of the function and the set is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. History of the function concept, Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable function, differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept. A f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Relation
In mathematics, a binary relation ''R'' ⊆ ''X''×''Y'' between two sets ''X'' and ''Y'' is total (or left total) if the source set ''X'' equals the domain . Conversely, ''R'' is called right total if ''Y'' equals the range . When ''f'': ''X'' → ''Y'' is a function, the domain of ''f'' is all of ''X'', hence ''f'' is a total relation. On the other hand, if ''f'' is a partial function, then the domain may be a proper subset of ''X'', in which case ''f'' is not a total relation. "A binary relation is said to be total with respect to a universe of discourse just in case everything in that universe of discourse stands in that relation to something else." from |
Partial Function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain of . If equals , that is, if is defined on every element in , then is said to be a total function. In other words, a partial function is a binary relation over two sets that associates to every element of the first set ''at most'' one element of the second set; it is thus a univalent relation. This generalizes the concept of a (total) function by not requiring ''every'' element of the first set to be associated to an element of the second set. A partial function is often used when its exact domain of definition is not known, or is difficult to specify. However, even when the exact domain of definition is known, partial functions are often used for simplicity or brevity. This is the case in calculus, where, for example, the quotien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Socratic Method
The Socratic method (also known as the method of Elenchus or Socratic debate) is a form of argumentative dialogue between individuals based on asking and answering questions. Socratic dialogues feature in many of the works of the ancient Greek philosopher Plato, where his teacher Socrates debates various philosophical issues with an "interlocutor" or "partner". In Plato's dialogue "Theaetetus (dialogue), Theaetetus", Socrates describes his method as a form of "midwifery" because it is employed to help his interlocutors develop their understanding in a way analogous to a Prenatal development, child developing in the womb. The Socratic method begins with commonly held beliefs and scrutinizes them by way of questioning to determine their internal consistency and their coherence with other beliefs and so to bring everyone closer to the truth. In modified forms, it is employed today in a variety of pedagogy, pedagogical contexts. Development In the second half of the 5th century B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Court Publishing Company
The Open Court Publishing Company is a publisher with offices in Chicago and LaSalle, Illinois. It is part of the Carus Publishing Company of Peru, Illinois. History Open Court was founded in 1887 by Edward C. Hegeler of the Matthiessen-Hegeler Zinc Company, at one time the largest producer of zinc in the United States. Hegeler intended for the firm to serve the purpose of discussing religious and psychological problems on the principle that the scientific world-conception should be applied to religion. Its first managing editor was Paul Carus, Hegeler's son-in-law through his marriage to engineer Mary Hegeler Carus.Fields 1992, pg. 138 For the first 80 years of its existence, the company had its offices in the Hegeler Carus Mansion. Open Court specializes in philosophy, science, and religion. It was one of the first academic presses in the country, as well as one of the first publishers of inexpensive editions of the classics. It also published the journals ''Open Court' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Presupposition
In linguistics and philosophy, a presupposition is an implicit assumption about the world or background belief relating to an utterance whose truth is taken for granted in discourse. Examples of presuppositions include: * ''Jane no longer writes fiction.'' ** Presupposition: Jane once wrote fiction. * ''Have you stopped eating meat?'' ** Presupposition: you had once eaten meat. * ''Have you talked to Hans?'' ** Presupposition: Hans exists. A presupposition is information that is linguistically presented as being mutually known or assumed by the speaker and addressee. This may be required for the utterance to be considered appropriate in context, but it is not uncommon for new information to be encoded in presuppositions without disrupting the flow of conversation (see accommodation below). A presupposition remains mutually known by the speaker and addressee whether the utterance is placed in the form of an assertion, denial, or question, and can be associated with a specific lexi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Question
A question is an utterance which serves as a request for information. Questions are sometimes distinguished from interrogatives, which are the grammar, grammatical forms, typically used to express them. Rhetorical questions, for instance, are interrogative in form but may not be considered wiktionary:bona fide, bona fide questions, as they are not expected to be answered. Questions come in a number of varieties. For instance; ''Polar questions'' are those such as the English language, English example "Is this a polar question?", which can be answered with yes and no, "yes" or "no". ''Alternative questions'' such as "Is this a polar question, or an alternative question?" present a list of possibilities to choose from. ''Open-ended question, Open questions'' such as "What kind of question is this?" allow many possible resolutions. Questions are widely studied in linguistics and philosophy of language. In the subfield of pragmatics, questions are regarded as illocutionary acts whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
