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Formal Theories
Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements (forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for formal events * Semi-formal attire, attire for semi-formal events * Informal attire, more controlled attire than casual but less than formal * Formal (university), official university dinner, ball or other event * School formal, official school dinner, ball or other event Logic and mathematics *Formal logic, or mathematical logic ** Informal logic, the complement, whose definition and scope is contentious *Formal fallacy, reasoning of invalid structure ** Informal fallacy, the complement *Informal mathematics, also called naïve mathematics *Formal cause, Aristotle's intrinsic, determining cause * Formal power series, a generalization of power series without requiring convergence, used in combinatorics *Formal calculation, a calculation which is systematic, but without a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies 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 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 theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Derivative
In mathematics, the formal derivative is an operation on elements of a polynomial ring or a ring of formal power series that mimics the form of the derivative from calculus. Though they appear similar, the algebraic advantage of a formal derivative is that it does not rely on the notion of a limit, which is in general impossible to define for a ring. Many of the properties of the derivative are true of the formal derivative, but some, especially those that make numerical statements, are not. Formal differentiation is used in algebra to test for multiple roots of a polynomial. Definition The definition of formal derivative is as follows: fix a ring ''R'' (not necessarily commutative) and let ''A'' = ''R'' 'x''be the ring of polynomials over ''R''. Then the formal derivative is an operation on elements of ''A'', where if :f(x)\,=\,a_n x^n + \cdots + a_1 x + a_0, then its formal derivative is :f'(x)\,=\,Df(x) = n a_n x^ + \cdots + 2 a_2 x + a_1, just as for polynomials over th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimethoxymethane
Dimethoxymethane, also called methylal, is a colorless flammable liquid with a low boiling point, low viscosity and excellent dissolving power. It has a chloroform-like odor and a pungent taste. It is the dimethyl acetal of formaldehyde. Dimethoxymethane is soluble in three parts water and miscible with most common organic solvents. Synthesis and structure It can be manufactured by oxidation of methanol or by the reaction of formaldehyde with methanol. In aqueous acid, it is hydrolyzed back to formaldehyde and methanol. Due to the anomeric effect, dimethoxymethane has a preference toward the ''gauche'' conformation with respect to each of the C–O bonds, instead of the ''anti'' conformation. Since there are two C–O bonds, the most stable conformation is ''gauche''-''gauche'', which is around 7 kcal/mol more stable than the ''anti''-''anti'' conformation, while the ''gauche''-''anti'' and ''anti-gauche'' are intermediate in energy. Since it is one of the smallest molecules ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ketal
In organic chemistry, an acetal is a functional group with the connectivity . Here, the R groups can be organic fragments (a carbon atom, with arbitrary other atoms attached to that) or hydrogen, while the R' groups must be organic fragments not hydrogen. The two R' groups can be equivalent to each other (a "symmetric acetal") or not (a "mixed acetal"). Acetals are formed from and convertible to aldehydes or ketones and have the same oxidation state at the central carbon, but have substantially different chemical stability and reactivity as compared to the analogous carbonyl compounds. The central carbon atom has four bonds to it, and is therefore saturated and has tetrahedral geometry. The term ketal is sometimes used to identify structures associated with ketones (both R groups organic fragments rather than hydrogen) rather than aldehydes and, historically, the term acetal was used specifically for the aldehyde-related cases (having at least one hydrogen in place of an R on t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acetal
In organic chemistry, an acetal is a functional group with the connectivity . Here, the R groups can be organic fragments (a carbon atom, with arbitrary other atoms attached to that) or hydrogen, while the R' groups must be organic fragments not hydrogen. The two R' groups can be equivalent to each other (a "symmetric acetal") or not (a "mixed acetal"). Acetals are formed from and convertible to aldehydes or ketones and have the same oxidation state at the central carbon, but have substantially different chemical stability and reactivity as compared to the analogous carbonyl compounds. The central carbon atom has four bonds to it, and is therefore saturated and has tetrahedral geometry. The term ketal is sometimes used to identify structures associated with ketones (both R groups organic fragments rather than hydrogen) rather than aldehydes and, historically, the term acetal was used specifically for the aldehyde-related cases (having at least one hydrogen in place of an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Concentration
Molar concentration (also called molarity, amount concentration or substance concentration) is a measure of the concentration of a chemical species, in particular of a solute in a Solution (chemistry), solution, in terms of amount of substance per unit volume of solution. In chemistry, the most commonly used unit for molarity is the number of Mole (unit), moles per liter, having the unit symbol mol/L or Mole (unit), mol/Decimetre, dm3 in SI unit. A solution with a concentration of 1 mol/L is said to be 1 molar, commonly designated as 1 M. Definition Molar concentration or molarity is most commonly expressed in units of moles of solute per litre of Solution (chemistry), solution. For use in broader applications, it is defined as amount of substance of solute per unit volume of solution, or per unit volume available to the species, represented by lowercase c: :c = \frac = \frac = \frac. Here, n is the amount of the solute in moles, N is the number of particle number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamic And Formal Equivalence
The terms dynamic equivalence and formal equivalence, coined by Eugene Nida, are associated with two dissimilar translation approaches that are employed to achieve different levels of literalness between the Translation#Source and target texts, source and target text, as evidenced in Bible translations, biblical translation. The two have been understood basically, with dynamic equivalence as sense-for-sense translation (translating the meanings of phrases or whole sentences) with readability in mind, and with formal equivalence as Literal translation, word-for-word translation (translating the meanings of words and phrases in a more literal way), keeping literal Fidelity#Translation, fidelity. Approaches to translation ''Formal equivalence approach'' tends to emphasize fidelity to the lexical details and grammatical structure of the original language, whereas dynamic equivalence tends to employ a more natural rendering but with less literal accuracy. According to Eugene Nid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Proof
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is not in general effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists. The concepts of Fitch-style proof, sequent calculus and natural deduction are generalizations of the concept of proof. The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof. For a well-formed formula to qualify as part of a proof, it must be the result of applying a rule of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T–V Distinction
The T–V distinction is the contextual use of different pronouns that exists in some languages and serves to convey formality or familiarity. Its name comes from the Latin pronouns '' tu'' and '' vos''. The distinction takes a number of forms and indicates varying levels of politeness, familiarity, courtesy, age or even insult toward the addressee. The field that studies and describes this phenomenon is sociolinguistics. Many languages lack this type of distinction, instead relying on other morphological or discourse features to convey formality. English historically contained the distinction, using the pronouns ''thou'' and ''you'', but the familiar ''thou'' largely disappeared from the era of Early Modern English onward, with the exception of a few dialects. Additionally, British commoners historically spoke to nobility and royalty using the third person rather than the second person, a practice that has fallen out of favour. English speakers today often employ semantic ana ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Grammar
In formal language theory, a grammar (when the context is not given, often called a formal grammar for clarity) describes how to form strings from a language's alphabet that are valid according to the language's syntax. A grammar does not describe the meaning of the strings or what can be done with them in whatever context—only their form. A formal grammar is defined as a set of production rules for such strings in a formal language. Formal language theory, the discipline that studies formal grammars and languages, is a branch of applied mathematics. Its applications are found in theoretical computer science, theoretical linguistics, formal semantics, mathematical logic, and other areas. A formal grammar is a set of rules for rewriting strings, along with a "start symbol" from which rewriting starts. Therefore, a grammar is usually thought of as a language generator. However, it can also sometimes be used as the basis for a "recognizer"—a function in computing that deter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of symbols, letters, or tokens that concatenate into strings of the language. Each string concatenated from symbols of this alphabet is called a word, and the words that belong to a particular formal language are sometimes called ''well-formed words'' or ''well-formed formulas''. A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar, which consists of its formation rules. In computer science, formal languages are used among others as the basis for defining the grammar of programming languages and formalized versions of subsets of natural languages in which the words of the language represent concepts that are associated with particular meanings or semantics. In computational complexity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal System
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system". In 1921, David Hilbert proposed to use such a system as the foundation for the knowledge in mathematics. A formal system may represent a well-defined abstraction, system of abstract thought. The term ''formalism'' is sometimes a rough synonym for ''formal system'', but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation. Background Each formal system is described by primitive Symbol (formal), symbols (which collectively form an Alphabet (computer science), alphabet) to finitely construct a formal language from a set of axioms through inferential rules of formation. The system thus consists of valid formulas built up through finite combinations of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
