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Vicious Circle Principle
The vicious circle principle is a principle that was endorsed by many predicativist mathematicians in the early 20th century to prevent contradictions. The principle states that no object or property may be introduced by a definition that depends on that object or property itself. In addition to ruling out definitions that are explicitly circular (like "an object has property ''P'' iff it is not next to anything that has property ''P''"), this principle rules out definitions that quantify over domains which include the entity being defined. Thus, it blocks Russell's paradox, which defines a set ''R'' that contains all sets which do not contain themselves. This definition is blocked because it defines a new set in terms of the totality of all sets, of which this new set would itself be a member. However, it also blocks one standard definition of the natural numbers. First, we define a property as being " hereditary" if, whenever a number ''n'' has the property, so does ''n''+ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Impredicativity
In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more commonly) another set that contains the thing being defined. There is no generally accepted precise definition of what it means to be predicative or impredicative. Authors have given different but related definitions. The opposite of impredicativity is predicativity, which essentially entails building stratified (or ramified) theories where quantification over a type at one 'level' results in types at a new, higher, level. A prototypical example is intuitionistic type theory, which retains ramification (without the explicit levels) so as to discard impredicativity. The 'levels' here correspond to the number of layers of dependency in a term definition. Russell's paradox is a famous example of an impredicative construction—namely the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type Theory
In mathematics and theoretical computer science, a type theory is the formal presentation of a specific type system. Type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of mathematics. Two influential type theories that have been proposed as foundations are: * Typed λ-calculus of Alonzo Church * Intuitionistic type theory of Per Martin-Löf Most computerized proof-writing systems use a type theory for their foundation. A common one is Thierry Coquand's Calculus of Inductive Constructions. History Type theory was created to avoid paradoxes in naive set theory and formal logic, such as Russell's paradox which demonstrates that, without proper axioms, it is possible to define the set of all sets that are not members of themselves; this set both contains itself and does not contain itself. Between 1902 and 1908, Bertrand Russell proposed various solutions to this problem. By 1908, Russell arrive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophical Logic
Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical logic in a wider sense as the study of the scope and nature of logic in general. In this sense, philosophical logic can be seen as identical to the philosophy of logic, which includes additional topics like how to define logic or a discussion of the fundamental concepts of logic. The current article treats philosophical logic in the narrow sense, in which it forms one field of inquiry within the philosophy of logic. An important issue for philosophical logic is the question of how to classify the great variety of non-classical logical systems, many of which are of rather recent origin. One form of classification often found in the literature is to distinguish between extended logics and deviant logics. Logic itself can be defined as t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concepts In Logic
A concept is an abstract idea that serves as a foundation for more concrete principles, thoughts, and beliefs. Concepts play an important role in all aspects of cognition. As such, concepts are studied within such disciplines as linguistics, psychology, and philosophy, and these disciplines are interested in the logical and psychological structure of concepts, and how they are put together to form thoughts and sentences. The study of concepts has served as an important flagship of an emerging interdisciplinary approach, cognitive science. In contemporary philosophy, three understandings of a concept prevail: * mental representations, such that a concept is an entity that exists in the mind (a mental object) * abilities peculiar to cognitive agents (mental states) * Fregean senses, abstract objects rather than a mental object or a mental state Concepts are classified into a hierarchy, higher levels of which are termed "superordinate" and lower levels termed "subordinate". A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Footnotes
In publishing, a note is a brief text in which the author comments on the subject and themes of the book and names supporting citations. In the editorial production of books and documents, typographically, a note is usually several lines of text at the bottom of the page, at the end of a chapter, at the end of a volume, or a house-style typographic usage throughout the text. Notes are usually identified with superscript numbers or a symbol.''The Oxford Companion to the English Language'' (1992) p. 709. Footnotes are informational notes located at the foot of the thematically relevant page, whilst endnotes are informational notes published at the end of a chapter, the end of a volume, or the conclusion of a multi-volume book. Unlike footnotes, which require manipulating the page design (text-block and page layouts) to accommodate the additional text, endnotes are advantageous to editorial production because the textual inclusion does not alter the design of the publication. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hofstadter's Law
Hofstadter's law is a self-referential adage, coined by Douglas Hofstadter in his book ''Gödel, Escher, Bach, Gödel, Escher, Bach: An Eternal Golden Braid'' (1979) to describe the widely experienced difficulty of accurately estimating the time it will take to complete tasks of substantial complexity:''Gödel, Escher, Bach: An Eternal Golden Braid''. 20th anniversary ed., 1999, p. 152. . The law is often cited by programmers in discussions of techniques to improve productivity, such as ''The Mythical Man-Month'' or extreme programming. History In 1979, Hofstadter introduced the law in connection with a discussion of Computer chess, chess-playing computers, which at the time were continually being beaten by Best chess players, top-level human players, despite outpacing humans in depth of recursion (computer science), analysis. Hofstadter wrote: In 1997, the chess computer Deep Blue (chess computer), Deep Blue became the first to beat a human champion by Deep Blue versus Garry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circular Definition
A circular definition is a type of definition that uses the term(s) being defined as part of the description or assumes that the term(s) being described are already known. There are several kinds of circular definition, and several ways of characterising the term: pragmatic, lexicographic and linguistic Linguistics is the scientific study of language. The areas of linguistic analysis are syntax (rules governing the structure of sentences), semantics (meaning), Morphology (linguistics), morphology (structure of words), phonetics (speech sounds .... Circular definitions are related to circular reasoning in that they both involve a self-referential approach. Circular definitions may be unhelpful if the audience must either already know the meaning of the key term, or if the term to be defined is used in the definition itself. In linguistics, a circular definition is a description of the meaning of a lexeme that is constructed using one or more synonymous lexemes that are al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-reference
Self-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields. In natural or formal languages, self-reference occurs when a sentence, idea or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding. In philosophy, self-reference also refers to the ability of a subject to speak of or refer to itself, that is, to have the kind of thought expressed by the first person nominative singular pronoun "I" in English. Self-reference is studied and has applications in mathematics, philosophy, computer programming, second-order cybernetics, and linguistics, as well as in humor. Self-referential statements are sometimes paradoxical, and can also be considered recursive. In logic, mathematics and computing In classical philosophy, paradoxes were created b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanford Encyclopedia Of Philosophy
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') is a freely available online philosophy resource published and maintained by Stanford University, encompassing both an online encyclopedia of philosophy and peer-reviewed original publication. Each entry is written and maintained by an expert in the field, including professors from many academic institutions worldwide. Authors contributing to the encyclopedia give Stanford University the permission to publish the articles, but retain the copyright to those articles. Approach and history As of August 5, 2022, the ''SEP'' has 1,774 published entries. Apart from its online status, the encyclopedia uses the traditional academic approach of most encyclopedias and academic journals to achieve quality by means of specialist authors selected by an editor or an editorial committee that is competent (although not necessarily considered specialists) in the field covered by the encyclopedia and peer review. The encyclopedia was created i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of Type Theory
The type theory was initially created to avoid paradoxes in a variety of formal logics and rewrite systems. Later, type theory referred to a class of formal systems, some of which can serve as alternatives to naive set theory as a foundation for all mathematics. It has been tied to formal mathematics since ''Principia Mathematica'' to today's proof assistants. 1900–1927 Origin of Russell's theory of types In a letter to Gottlob Frege (1902), Bertrand Russell announced his discovery of the paradox in Frege's Begriffsschrift. Frege promptly responded, acknowledging the problem and proposing a solution in a technical discussion of "levels". To quote Frege: Incidentally, it seems to me that the expression "a predicate is predicated of itself" is not exact. A predicate is as a rule a first-level function, and this function requires an object as argument and cannot have itself as argument (subject). Therefore I would prefer to say "a concept is predicated of its own extension" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russell's Paradox
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. According to the unrestricted comprehension principle, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. Let ''R'' be the set of all sets that are not members of themselves. (This set is sometimes called "the Russell set".) If ''R'' is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols: : Let R = \. Then R \in R \iff R \not \in R. Russell also showed that a version of the paradox co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of Type Theory
The type theory was initially created to avoid paradoxes in a variety of formal logics and rewrite systems. Later, type theory referred to a class of formal systems, some of which can serve as alternatives to naive set theory as a foundation for all mathematics. It has been tied to formal mathematics since ''Principia Mathematica'' to today's proof assistants. 1900–1927 Origin of Russell's theory of types In a letter to Gottlob Frege (1902), Bertrand Russell announced his discovery of the paradox in Frege's Begriffsschrift. Frege promptly responded, acknowledging the problem and proposing a solution in a technical discussion of "levels". To quote Frege: Incidentally, it seems to me that the expression "a predicate is predicated of itself" is not exact. A predicate is as a rule a first-level function, and this function requires an object as argument and cannot have itself as argument (subject). Therefore I would prefer to say "a concept is predicated of its own extension" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
