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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] [Amazon] |
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] [Amazon] |
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] [Amazon] |
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] [Amazon] |
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] [Amazon] |
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] [Amazon] |
