|
Modal Logic
Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality, causation. For instance, in epistemic modal logic, the well-formed_formula, formula \Box P can be used to represent the statement that P is known. In deontic modal logic, that same formula can represent that P is a moral obligation. Modal logic considers the inferences that modal statements give rise to. For instance, most epistemic modal logics treat the formula \Box P \rightarrow P as a Tautology_(logic), tautology, representing the principle that only true statements can count as knowledge. However, this formula is not a tautology in deontic modal logic, since what ought to be true can be false. Modal logics are formal systems that include unary operation, unary operators such as \Diamond and \Box, representing possibility and necessi ... [...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 study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiomatic System
In mathematics and logic, an axiomatic system is a set of formal statements (i.e. axioms) used to logically derive other statements such as lemmas or theorems. A proof within an axiom system is a sequence of deductive steps that establishes a new statement as a consequence of the axioms. An axiom system is called complete with respect to a property if every formula with the property can be derived using the axioms. The more general term theory is at times used to refer to an axiomatic system and all its derived theorems. In its pure form, an axiom system is effectively a syntactic construct and does not by itself refer to (or depend on) a formal structure, although axioms are often defined for that purpose. The more modern field of model theory refers to mathematical structures. The relationship between an axiom systems and the models that correspond to it is often a major issue of interest. Properties Four typical properties of an axiom system are consistency, relativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinism
Determinism is the Metaphysics, metaphysical view that all events within the universe (or multiverse) can occur only in one possible way. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and considerations. Like Eternalism (philosophy of time), eternalism, determinism focuses on particular events rather than the future as a concept. Determinism is often contrasted with free will, although some philosophers claim that the two are compatibilism, compatible. A more extreme antonym of determinism is indeterminism, or the view that events are not deterministically caused but rather occur due to random chance. Historically, debates about determinism have involved many philosophical positions and given rise to multiple varieties or interpretations of determinism. One topic of debate concerns the scope of determined systems. Some philosophers have maintained that the entire universe is a single determinate system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator (mathematics)
In mathematics, an operator is generally a Map (mathematics), mapping or function (mathematics), function that acts on elements of a space (mathematics), space to produce elements of another space (possibly and sometimes required to be the same space). There is no general definition of an ''operator'', but the term is often used in place of ''function'' when the domain of a function, domain is a set of functions or other structured objects. Also, the domain of an operator is often difficult to characterize explicitly (for example in the case of an integral operator), and may be extended so as to act on related objects (an operator that acts on functions may act also on differential equations whose solutions are functions that satisfy the equation). (see Operator (physics) for other examples) The most basic operators are linear maps, which act on vector spaces. Linear operators refer to linear maps whose domain and range are the same space, for example from \mathbb^n to \mathbb^n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Social Epistemology
Social epistemology refers to a broad set of approaches that can be taken in epistemology (the study of knowledge) that construes human knowledge as a collective achievement. Another way of characterizing social epistemology is as the evaluation of the social dimensions of knowledge or information. As a field of inquiry in analytic philosophy, social epistemology deals with questions about knowledge in social contexts, meaning those in which knowledge attributions cannot be explained by examining individuals in isolation from one another. The most common topics discussed in contemporary social epistemology are testimony (e.g. "When does a belief that x is true which resulted from being told 'x is true' constitute knowledge?"), peer disagreement (e.g. "When and how should I revise my beliefs in light of other people holding beliefs that contradict mine?"), and group epistemology (e.g. "What does it mean to attribute knowledge to groups rather than individuals, and when are such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiverse (set Theory)
In mathematical set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ..., the multiverse view is that there are many models of set theory, but no "absolute", "canonical" or "true" model. The various models are all equally valid or true, though some may be more useful or attractive than others. The opposite view is the "universe" view of set theory in which all sets are contained in some single ultimate model. The collection of countable transitive models of ZFC (in some universe) is called the hyperverse and is very similar to the "multiverse". A typical difference between the universe and multiverse views is the attitude to the continuum hypothesis. In the universe view the continuum hypothesis is a meaningful question that is either true or false though we have not yet b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Web Design
Web design encompasses many different skills and disciplines in the production and maintenance of websites. The different areas of web design include web graphic design; user interface design (UI design); authoring, including standardised code and proprietary software; user experience design (UX design); and search engine optimization. Often many individuals will work in teams covering different aspects of the design process, although some designers will cover them all. The term "web design" is normally used to describe the design process relating to the front-end (client side) design of a website including writing markup. Web design partially overlaps web engineering in the broader scope of web development. Web designers are expected to have an awareness of usability and be up to date with web accessibility guidelines. History 1988–2001 Although web design has a fairly recent history, it can be linked to other areas such as graphic design, user experience, and multime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legal Theory
Jurisprudence, also known as theory of law or philosophy of law, is the examination in a general perspective of what law is and what it ought to be. It investigates issues such as the definition of law; legal validity; legal norms and values; and the relationship between law and other fields of study, including economics, ethics, history, sociology, and political philosophy. Modern jurisprudence began in the 18th century and was based on the first principles of natural law, civil law, and the law of nations. Contemporary philosophy of law addresses problems internal to law and legal systems and problems of law as a social institution that relates to the larger political and social context in which it exists. Jurisprudence can be divided into categories both by the type of question scholars seek to answer and by the theories of jurisprudence, or schools of thought, regarding how those questions are best answered: * Natural law holds that there are rational objective limits ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moral Theory
Ethics is the philosophy, philosophical study of Morality, moral phenomena. Also called moral philosophy, it investigates Normativity, normative questions about what people ought to do or which behavior is morally right. Its main branches include normative ethics, applied ethics, and metaethics. Normative ethics aims to find general principles that govern how people should act. Applied ethics examines concrete ethical problems in real-life situations, such as abortion, treatment of animals, and Business ethics, business practices. Metaethics explores the underlying assumptions and concepts of ethics. It asks whether there are objective moral facts, how moral knowledge is possible, and how moral judgments motivate people. Influential normative theories are consequentialism, deontology, and virtue ethics. According to consequentialists, an act is right if it leads to the best consequences. Deontologists focus on acts themselves, saying that they must adhere to Duty, duties, like t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed two-person zero-sum games, in which a participant's gains or losses are exactly balanced by the losses and gains of the other participant. In the 1950s, it was extended to the study of non zero-sum games, and was eventually applied to a wide range of Human behavior, behavioral relations. It is now an umbrella term for the science of rational Decision-making, decision making in humans, animals, and computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighborhood Semantics
Neighborhood semantics, also known as Scott–Montague semantics, is a formal semantics for modal logics. It is a generalization, developed independently by Dana Scott and Richard Montague, of the more widely known relational semantics for modal logic. Whereas a relational frame \langle W,R\rangle consists of a set ''W'' of worlds (or states) and an accessibility relation ''R'' intended to indicate which worlds are alternatives to (or, accessible from) others, a neighborhood frame \langle W,N\rangle still has a set ''W'' of worlds, but has instead of an accessibility relation a ''neighborhood function'' : N : W \to 2^ that assigns to each element of ''W'' a set of subsets of ''W''. Intuitively, each family of subsets assigned to a world are the propositions necessary at that world, where 'proposition' is defined as a subset of ''W'' (i.e. the set of worlds at which the proposition is true). Specifically, if ''M'' is a model on the frame, then : M,w\models\square \varphi \L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Torsion (mechanics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a Set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of List of continuity-related mathematical topics, continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and Homotopy, homotopies. A property that is invariant under such deformations is a to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
