In the formal languages A formal language is a set of words, i.e. finite strings of letters, symbols, or tokens. The set from which these letters are taken is called the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar ; accordingly, words that belong to a formal language are sometimes called well-formed words ( used in mathematical logic Mathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the and computer science Computer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems. It is frequently described as the systematic study of algorithmic processes that create, describe and transform information. According to Peter J. Denning, the, a well-formed formula or simply formula[2] (often abbreviated wff, pronounced "wiff" or "wuff") is an idea In the most narrow sense, an idea is just whatever is before the mind when one thinks. Very often, ideas are construed as representational images; i.e. images of some object. In other contexts, ideas are taken to be concepts, although abstract concepts do not necessarily appear as images. Many philosophers consider ideas to be a fundamental, abstraction Abstraction is the process or result of generalization by reducing the information content of a concept or an observable phenomenon, typically to retain only information which is relevant for a particular purpose. For example, abstracting a leather soccer ball to a ball retains only the information on general ball attributes and behaviour or concept There are two prevailing theories in contemporary philosophy which attempt to explain the nature of concepts . The representational theory of mind proposes that concepts are mental representations, while the semantic theory of concepts (originating with Frege's distinction between concept and object) holds that they are abstract objects. Ideas are which is expressed using the symbols A symbol is an idea, abstraction or concept, tokens of which may be marks or a configuration of marks which form a particular pattern. Although the term "symbol" in common use refers at some times to the idea being symbolized, and at other times to the marks on a piece of paper or chalkboard which are being used to express that idea; in and formation rules In mathematical logic, formation rules are rules for describing which strings of symbols formed from the alphabet of a formal language are syntactically valid within the language. A grammar only addresses the location and manipulation of the strings of the language. It does not describe anything else about a language, such as its semantics (also called the formal grammar A formal grammar is a set of rules for forming strings in a formal language. These rules that make up the grammar describe how to form strings from the language's alphabet that are valid according to the language's syntax. A grammar does not describe the meaning of the strings—only their location and the ways that they can be manipulated) of a particular formal language. To say that a string In mathematics, a string is a sequence of symbols that are chosen from a set or alphabet of symbols is a wff with respect to a given formal grammar is equivalent to saying that belongs to the language generated by . A formal language can be identified with the set of its wffs.
Although the term "well-formed formula" is commonly used to refer to the written marks, for instance, on a piece of paper or chalkboard which are being used to express an idea; it is more precisely understood as the idea being expressed and the marks as a token In philosophy and knowledge representation, the type-token distinction is a distinction that separates an abstract concept from the objects which are particular instances of the concept. For example, the particular bicycle in your garage is a token of the type of thing known as "The bicycle." Whereas, the bicycle in your garage is in a instance of the well formed formula. Two different strings of marks may be tokens of the same well-formed formula. This is to say that there may be many different formulations In logic and mathematics, a formulation is a particular sequence of symbols which together represent some concept. For any given concept, there may be an infinite number of possible formulations. There also may be formulations of the same concept in different languages of the same the idea.
It is not necessary for the existence of a well-formed formula that there be any actual tokens of it. Formal languages may have an infinite number of well-formed formula, regardless of whether there actually exist any token instances of them.
Well-formed formulas are quite often interpreted An interpretation is an assignment of "meaning" to the symbols of a language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any "meaning" until they are given some interpretation. The general study of interpretations of formal as propositions A proposition is a sentence expressing something true or false. In philosophy, particularly in logic, a proposition is identified ontologically as an idea, concept, or abstraction whose token instances are patterns of symbols, marks, sounds, or strings of words. Propositions are considered to be syntactic entities and also truthbearers (as, for instance, in propositional logic In logic and mathematics, a propositional calculus or logic is a formal system in which formulae representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows certain formulae to be established as theorems). However wffs are syntactic entities In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the semantics of a language which is concerned with its meaning, and as such must be specified in a formal language without regard to any interpretation An interpretation is an assignment of "meaning" to the symbols of a language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any "meaning" until they are given some interpretation. The general study of interpretations of formal of them. An interpreted well-formed formula may be the name A name is a label for a noun, normally used to distinguish one from another. Names can identify a class or category of things, or a single thing, either uniquely, or within a given context. A personal name identifies a specific unique and identifiable individual person. The name of a specific entity is sometimes called a proper name and is a of something, an adjective In grammar, an adjective is a word whose main syntactic role is to modify a noun or pronoun, giving more information about the noun or pronoun's referent. Collectively, adjectives form one of the traditional English eight parts of speech, though linguists today distinguish adjectives from words such as determiners that also used to be considered, an adverb An adverb is a part of speech. It is any word that modifies any other part of language: verbs, adjectives , clauses, sentences and other adverbs, except for nouns; modifiers of nouns are primarily determiners and adjectives, a preposition In grammar, a preposition is a part of speech that introduces a prepositional phrase. For example, in the sentence "The cat sleeps on the sofa", the word "on" is a preposition, introducing the prepositional phrase "on the sofa". In English, the most used prepositions are "of", "to", "in",, a phrase In grammar, a phrase is a group of words functioning as a single unit in the syntax of a sentence, a clause In grammar, a clause is a pair or group of words that consist of a subject and a predicate, although in some languages and some types of clauses, the subject may not appear explicitly as a noun phrase. It may instead be marked on the verb . The most basic kind of sentence consists of a single clause; more complicated sentences may contain multiple, an imperative sentence The English imperative is formed simply by using the bare infinitive form of the verb. Be is the only verb whose infinitive form is in different from the second-person present indicative form. The subject of the sentence can only be you . Other languages such as Latin, French and German have several inflected imperative forms, which can vary, a string In mathematics, a string is a sequence of symbols that are chosen from a set or alphabet of sentences, a string of names, etcetera. A well-formed formula may even turn out to be nonsense Nonsense (pronounced \ˈnän-ˌsens, ˈnän(t)-sən(t)s\) is a verbal communication or written text which resembles a human language or other symbolic system, but which lacks any coherent meaning, if the symbols of the language are specified so that it does. Furthermore, a well-formed formula need not be given any interpretation.
The set A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In of well-formed formulas of a particular formal language is determined by a fiat of its creator, who simply lays down what things are to be wffs of his language. Usually this is done by specifying a set of symbols In computer science and mathematical logic, an alphabet is a, usually finite, set of symbols or letters, e.g. characters or digits. The most common alphabet is {0,1}, the binary alphabet. A finite string is a finite sequence of letters from an alphabet; for instance a binary string is a string drawn from the alphabet {0,1}. An infinite sequence of, and a set of formation rules In mathematical logic, formation rules are rules for describing which strings of symbols formed from the alphabet of a formal language are syntactically valid within the language. A grammar only addresses the location and manipulation of the strings of the language. It does not describe anything else about a language, such as its semantics.
A key use of wffs is in propositional logic In logic and mathematics, a propositional calculus or logic is a formal system in which formulae representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows certain formulae to be established as theorems and predicate logics In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulas contain variables which can be quantified. Two common quantifiers are the existential ∃ and universal such as first-order logic First-order logic is a formal logic used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, and predicate logic. First-order logic is distinguished from propositional logic by its use of quantifiers; each interpretation of first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where substitution may take place. The idea is related to a placeholder , or a wildcard character that stands for an unspecified symbol in φ have been instantiated.
In formal logic, proofs In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception. An unproved proposition that is believed to can be represented by sequences of wffs with certain properties, and the final wff in the sequence is what is proven. This final wff is called a theorem In mathematics, a theorem is a statement which has been proved on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. In formal mathematical logic, the concept of a theorem may be taken to mean a formula that can be derived according to the derivation rules of a fixed formal when it plays a significant role in the theory being developed, or a lemma In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself. A good stepping stone leads to many others, so some of the most powerful results in mathematics are known as lemmata, such as Bézout's lemma, Urysohn's lemma, Dehn's lemma, Fatou's lemma, Gauss's lemma, when it plays an accessory role in the proof of a theorem.
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ONE of Britain s leading universities has rejected an independent school pupil with four A grades at A level after applying a mathematical formula that gives an automatic advantage to pupils from poorly performing schools
biswajit
Mon, 17 Aug 2009 14:01:00 GM
I never understood the . logic. of what I need to learn in that time limit. I never got my calculation right. You have the . formula. , you have the calculator. Give that to any sixth class student and they will find the answer. ...
Q. Logic and mathematical formulas please? Thanks everybody, especially kichka_2002, Mal*, wisecrack, bluehoney, maxvijay, and carlosg I really appreciate your help for the details and the logic. I wish I could choose all of you as the best answer.
Asked by Ashley O - Tue Aug 8 07:45:08 2006 - - 18 Answers - 0 Comments
A. 35*8=280 x+x+8=280 2x=272 x=136 x+8=144 or alternativel\y: x+y=280 x-y=8 add the two above and you get: 2x=288 x=144 => y=144-8=136
Answered by kichka_2002 - Tue Aug 8 07:51:24 2006
