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Intensional Equality
In logic, extensional and intensional definitions are two key ways in which the objects, concepts, or referents a term refers to can be defined. They give meaning or denotation to a term. Intensional definition An intensional definition gives meaning to a term by specifying necessary and sufficient conditions for when the term should be used. In the case of nouns, this is equivalent to specifying the properties that an object needs to have in order to be counted as a referent of the term. For example, an intensional definition of the word "bachelor" is "unmarried man". This definition is valid because being an unmarried man is both a necessary condition and a sufficient condition for being a bachelor: it is necessary because one cannot be a bachelor without being an unmarried man, and it is sufficient because any unmarried man is a bachelor.Cook, Roy T. "Intensional Definition". In ''A Dictionary of Philosophical Logic''. Edinburgh: Edinburgh University Press, 2009. 155. This i ... [...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 science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usually un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. History The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''. Georg Cantor, one of the founders of set theory, gave the following defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extensional Context
In any of several fields of study that treat the use of signs — for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language — an extensional context (or transparent context) is a syntactic environment in which a sub-sentential expression ''e'' can be replaced by an expression with the same extension and without affecting the truth-value of the sentence as a whole. Extensional contexts are contrasted with opaque contexts where truth-preserving substitutions are not possible. Take the case of Clark Kent, who is secretly Superman. Suppose that Lois Lane fell out of a window and Superman caught her. Thus the sentence "Superman caught Lois Lane" is true. Because this sentence is an extensional context, the sentence "Clark Kent caught Lois Lane" is also true. Anybody that Superman caught, Clark Kent caught. In opposition to extensional contexts are intensional contexts (which can involve modal operators and modal logic), where terms cannot be s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extension (predicate Logic)
The extension of a predicatea truth-valued functionis the set of tuples of values that, used as arguments, satisfy the predicate. Such a set of tuples is a relation. Examples For example, the statement "''d2'' is the weekday following ''d1''" can be seen as a truth function associating to each tuple (''d2'', ''d1'') the value ''true'' or ''false''. The extension of this truth function is, by convention, the set of all such tuples associated with the value ''true'', i.e. By examining this extension we can conclude that "Tuesday is the weekday following Saturday" (for example) is false. Using set-builder notation, the extension of the ''n''-ary predicate \Phi can be written as :\\,. Relationship with characteristic function If the values 0 and 1 in the range of a characteristic function are identified with the values false and true, respectivelymaking the characteristic function a predicate, then for all relations ''R'' and predicates \Phi the following two statements are equiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comprehension (logic)
In logic, the comprehension of an object is the totality of intensions, that is, attributes, characters, marks, properties, or qualities, that the object possesses, or else the totality of intensions that are pertinent to the context of a given discussion. This is the correct technical term for the whole collection of intensions of an object, but it is common in less technical usage to see 'intension' used for both the composite and the primitive ideas. See also * Extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ... * Extensional definition * Intension * Intensional definition Concepts in logic Definition {{Logic-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rudolf Carnap
Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism. He is considered "one of the giants among twentieth-century philosophers." Biography Carnap's father had risen from being a poor ribbon-weaver to be the owner of a ribbon-making factory. His mother came from an academic family; her father was an educational reformer and her oldest brother was the archaeologist Wilhelm Dörpfeld. As a ten-year-old, Carnap accompanied Wilhelm Dörpfeld on an expedition to Greece. Carnap was raised in a profoundly religious Protestant family, but later became an atheist. He began his formal education at the Barmen Gymnasium and the Gymnasium in Jena. From 1910 to 1914, he attended the University of Jena, intending to write a thesis in physics. He also intently studied Immanuel Kant's '' Critique of Pure Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constance Jones
Emily Elizabeth Constance Jones (19 February 1848 – 9 April 1922) known as Constance Jones or E.E. Constance Jones, was an English philosopher and educator. She worked in logic and ethics. Life and career Emily Elizabeth Constance Jones was born at Langstone Court, Llangarron, Herefordshire, to John Jones and his wife, Emily, daughter of Thomas Oakley JP, of Monmouthshire. She was the eldest of ten children. Constance was mostly tutored at home. She spent her early teenage years with her family in Cape Town, South Africa, and when they returned to England in 1865 she attended a small school, Miss Robinson's, in Cheltenham, for a year. She was coached for the entrance examination for Girton College, Cambridge by Miss Alice Grüner, a former student of Newnham at her home in Sydenham, Kent. She went up to Girton in 1875 where, prompted by having read Henry Fawcett's ''Political Economy'' (1863) and Mill's ''Logic'' (1843), she chose the Moral Sciences Tripos. However, she almos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intension
In any of several fields of study that treat the use of signs — for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language — an intension is any property or quality connoted by a word, phrase, or another symbol. In the case of a word, the word's definition often implies an intension. For instance, the intensions of the word ''plant'' include properties such as "being composed of cellulose", "alive", and "organism", among others. A '' comprehension'' is the collection of all such intensions. Overview The meaning of a word can be thought of as the bond between the ''idea the word means'' and the ''physical form of the word''. Swiss linguist Ferdinand de Saussure (1857–1913) contrasts three concepts: # the ''signifier'' – the "sound image" or the string of letters on a page that one recognizes as the form of a sign # the ''signified'' – the meaning, the concept or idea that a sign expresses or evokes # the ''referent'' – the actu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intensional Definition
In logic, extensional and intensional definitions are two key ways in which the objects, concepts, or referents a term refers to can be defined. They give meaning or denotation to a term. Intensional definition An intensional definition gives meaning to a term by specifying necessary and sufficient conditions for when the term should be used. In the case of nouns, this is equivalent to specifying the properties that an object needs to have in order to be counted as a referent of the term. For example, an intensional definition of the word "bachelor" is "unmarried man". This definition is valid because being an unmarried man is both a necessary condition and a sufficient condition for being a bachelor: it is necessary because one cannot be a bachelor without being an unmarried man, and it is sufficient because any unmarried man is a bachelor.Cook, Roy T. "Intensional Definition". In ''A Dictionary of Philosophical Logic''. Edinburgh: Edinburgh University Press, 2009. 155. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ostensive Definition
An ostensive definition conveys the meaning of a term by pointing out examples. This type of definition is often used where the term is difficult to define verbally, either because the words will not be understood (as with children and new speakers of a language) or because of the nature of the term (such as colors or sensations). It is usually accompanied with a gesture pointing to the object serving as an example, and for this reason is also often referred to as " definition by pointing". Overview An ostensive definition assumes the questioner has sufficient understanding to recognize the type of information being given. Ludwig Wittgenstein writes: So one might say: the ostensive definition explains the use—the meaning—of the word when the overall role of the word in language is clear. Thus if I know that someone means to explain a colour-word to me the ostensive definition "That is called 'sepia' " will help me to understand the word.... One has already to know (or be ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enumerative Definition
An enumerative definition of a concept or term is a special type of extensional definition that gives an explicit and exhaustive listing of all the objects that fall under the concept or term in question. Enumerative definitions are only possible for finite sets and only practical for relatively small sets. An example of an enumerative definition for the set ; extant monotreme species (for which the intensional definition is ''species of currently-living mammals that lay eggs'') would be : platypuses : echidnae: :: short-beaked echidna :: long-beaked echidnae: ::: Sir David's long-beaked echidna ::: eastern long-beaked echidna ::: western long-beaked echidna See also * Definition * Extension * Extensional definition * Set notation * Enumeration An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Object (philosophy)
An object is a philosophical term often used in contrast to the term '' subject''. A subject is an observer and an object is a thing observed. For modern philosophers like Descartes, consciousness is a state of cognition that includes the subject—which can never be doubted as only it can be the one who doubts—and some object(s) that may be considered as not having real or full existence or value independent of the subject who observes it. Metaphysical frameworks also differ in whether they consider objects existing independently of their properties and, if so, in what way. The pragmatist Charles S. Peirce defines the broad notion of an object as anything that we can think or talk about. In a general sense it is any entity: the pyramids, gods, Socrates, Alpha Centauri, the number seven, a disbelief in predestination or the fear of cats. In a strict sense it refers to any definite being. A related notion is objecthood. Objecthood is the state of being an object. One approac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
