|
Categorical Formulation Of Order Theory
Categorical may refer to: * Categorical imperative, a concept in philosophy developed by Immanuel Kant * Categorical theory, in mathematical logic * Morley's categoricity theorem, a mathematical theorem in model theory * Categorical data analysis * Categorical distribution, a probability distribution * Categorical logic, a branch of category theory within mathematics with notable connections to theoretical computer science * Categorical syllogism, a kind of logical argument * Categorical proposition, a part of deductive reasoning * Categorization * Categorical perception * Category theory in mathematics ** Categorical set theory ** Categorical probability * Recursive categorical syntax in linguistics See also *Category (other) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Imperative
The categorical imperative () is the central philosophical concept in the deontological Kantian ethics, moral philosophy of Immanuel Kant. Introduced in Kant's 1785 ''Groundwork of the Metaphysics of Morals'', it is a way of evaluating motivations for action. It is best known in its original formulation: "Act only according to that maxim (philosophy), maxim whereby you can at the same time will that it should become a universal law."It is standard to also reference the ''Akademie Ausgabe'' of Kant's works. The ''Groundwork'' occurs in the fourth volume. Citations throughout this article follow the format 4:x. For example, the above citation is taken from 4:421. According to Kant, rational being (Kantian ethics), rational beings occupy a special place in creation, and morality can be summed up in an imperative, or ultimate commandment of reason, from which all duties and obligations derive. He defines an ''imperative'' as any proposition declaring a certain action (or inaction) t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Theory
In mathematical logic, a theory is categorical if it has exactly one model ( up to isomorphism). Such a theory can be viewed as ''defining'' its model, uniquely characterizing the model's structure. In first-order logic, only theories with a finite model can be categorical. Higher-order logic contains categorical theories with an infinite model. For example, the second-order Peano axioms are categorical, having a unique model whose domain is the set of natural numbers \mathbb. In model theory, the notion of a categorical theory is refined with respect to cardinality. A theory is -categorical (or categorical in ) if it has exactly one model of cardinality up to isomorphism. Morley's categoricity theorem is a theorem of stating that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities. extended Morley's theorem to uncountable languages: if the language has cardinality and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Data
In statistics, a categorical variable (also called qualitative variable) is a variable (research), variable that can take on one of a limited, and usually fixed, number of possible values, assigning each individual or other unit of observation to a particular group or nominal category on the basis of some qualitative property. In computer science and some branches of mathematics, categorical variables are referred to as enumerations or enumerated types. Commonly (though not in this article), each of the possible values of a categorical variable is referred to as a level. The probability distribution associated with a random variable, random categorical variable is called a categorical distribution. Categorical data is the statistical data type consisting of categorical variables or of data that has been converted into that form, for example as grouped data. More specifically, categorical data may derive from observations made of qualitative data that are summarised as counts or cros ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Distribution
In probability theory and statistics, a categorical distribution (also called a generalized Bernoulli distribution, multinoulli distribution) is a discrete probability distribution that describes the possible results of a random variable that can take on one of ''K'' possible categories, with the probability of each category separately specified. There is no innate underlying ordering of these outcomes, but numerical labels are often attached for convenience in describing the distribution, (e.g. 1 to ''K''). The ''K''-dimensional categorical distribution is the most general distribution over a ''K''-way event; any other discrete distribution over a size-''K'' sample space is a special case. The parameters specifying the probabilities of each possible outcome are constrained only by the fact that each must be in the range 0 to 1, and all must sum to 1. The categorical distribution is the generalization of the Bernoulli distribution for a categorical random variable, i.e. for a dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Logic
__NOTOC__ Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category (mathematics), category, and an Interpretation (logic), interpretation by a functor. The categorical framework provides a rich conceptual background for logical and type theory, type-theoretic constructions. The subject has been recognisable in these terms since around 1970. Overview There are three important themes in the categorical approach to logic: ;Categorical semantics: Categorical logic introduces the notion of ''structure valued in a category'' C with the classical model theory, model theoretic notion of a structure appearing in the particular case where C is the Category of sets, category of sets and functions. This notion has proven useful when the set-theoreti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Syllogism
A syllogism (, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. In its earliest form (defined by Aristotle in his 350 BC book '' Prior Analytics''), a deductive syllogism arises when two true premises (propositions or statements) validly imply a conclusion, or the main point that the argument aims to get across. For example, knowing that all men are mortal (major premise), and that Socrates is a man (minor premise), we may validly conclude that Socrates is mortal. Syllogistic arguments are usually represented in a three-line form: All men are mortal. Socrates is a man. Therefore, Socrates is mortal.In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism. From the Middle Ages onwards, ''categorical syllogism'' and ''syllogism'' were usually used interchangeably. This article is concerne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Proposition
In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the ''subject term'') are included in another (the ''predicate term''). The study of arguments using categorical statements (i.e., syllogisms) forms an important branch of deductive reasoning that began with the Ancient Greeks. The Ancient Greeks such as Aristotle identified four primary distinct types of categorical proposition and gave them standard forms (now often called ''A'', ''E'', ''I'', and ''O''). If, abstractly, the subject category is named ''S'' and the predicate category is named ''P'', the four standard forms are: * All ''S'' are ''P''. (''A'' form) * No ''S'' are ''P''. (''E'' form) * Some ''S'' are ''P''. (''I'' form) * Some ''S'' are not ''P''. (''O'' form) A large number of sentences may be translated into one of these canonical forms while retaining all or most of the original meaning of the sentence. Greek inves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorization
Classification is the activity of assigning objects to some pre-existing classes or categories. This is distinct from the task of establishing the classes themselves (for example through cluster analysis). Examples include diagnostic tests, identifying spam emails and deciding whether to give someone a driving license. As well as 'category', synonyms or near-synonyms for 'class' include 'type', 'species', 'order', 'concept', 'taxon', 'group', 'identification' and 'division'. The meaning of the word 'classification' (and its synonyms) may take on one of several related meanings. It may encompass both classification and the creation of classes, as for example in 'the task of categorizing pages in Wikipedia'; this overall activity is listed under taxonomy. It may refer exclusively to the underlying scheme of classes (which otherwise may be called a taxonomy). Or it may refer to the label given to an object by the classifier. Classification is a part of many different kinds of acti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Perception
Categorical perception is a phenomenon of perception of distinct categories when there is gradual change in a variable along a continuum. It was originally observed for auditory stimuli but now found to be applicable to other perceptual modalities. Motor theory of speech perception If one analyzes the sound spectrogram of aand a for example, and can be visualized as lying somewhere on an acoustic continuum based on their VOT (voice onset time). It is possible to construct a continuum of some intermediate tokens lying between the and endpoints by gradually decreasing the voice onset time. Alvin Liberman and colleagues (he did not talk about voice onset time in that paper) reported that when people listen to sounds that vary along the voicing continuum, they perceive only /ba/s and /pa/s, nothing in between. This effect—in which a perceived quality jumps abruptly from one category to another at a certain point along a continuum, instead of changing gradually—he dubbed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Set Theory
Categorical set theory is any one of several versions of set theory developed from or treated in the context of mathematical category theory. See also * Categorical logic __NOTOC__ Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science. In broad terms, cate ... References * * * * * * * * * * External links * Category theory Set theory Formal methods Categorical logic {{cattheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
