Normal modal logic

In logic, a normal modal logic is a set L of modal formulas such that L contains:

• All propositional tautologies;
• All instances of the Kripke schema: $\Box(A\to B)\to(\Box A\to\Box B)$

and it is closed under:

• Detachment rule (Modus Ponens): $A\to B, A \vdash B$;
• Necessitation rule: $\vdash A$ implies $\vdash\Box A$.

The smallest logic satisfying the above conditions is called K. Most modal logics commonly used nowadays (in terms of having philosophical motivations), e.g. C. I. Lewis's S4 and S5, are extensions of K. However a number of deontic and epistemic logics, for example, are non-normal, often because they give up the Kripke schema.

This logic-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Modal logic Logic stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем решить контрольную работу Anatoly Guzhvin Sound system (DJ) Look at other dictionaries: Modal logic — is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals words that express modalities qualify a statement. For example, the statement John is happy might be qualified by… …   Wikipedia Classical modal logic — In modal logic, a classical modal logic L is any modal logic containing (as axiom or theorem) the duality of the modal operators which is also closed under the rule Alternatively one can give a dual definition of L by which L is classical iff it… …   Wikipedia S5 (modal logic) — In logic and philosophy, S5 is one of five systems of modal logic proposed by Clarence Irving Lewis and Cooper Harold Langford in their 1932 book Symbolic Logic . It is a normal modal logic, and one of the oldest systems of modal logic of any… …   Wikipedia Regular modal logic — In modal logic, a regular modal logic L is a modal logic closed underDiamond A equiv lnotBoxlnot Aand the rule(Aland B) o C vdash (Box AlandBox B) oBox C.Every regular modal logic is classical, and every normal modal logic is regular and hence… …   Wikipedia Modal companion — In logic, a modal companion of a superintuitionistic (intermediate) logic L is a normal modal logic which interprets L by a certain canonical translation, described below. Modal companions share various properties of the original intermediate… …   Wikipedia logic, philosophy of — Philosophical study of the nature and scope of logic. Examples of questions raised in the philosophy of logic are: In virtue of what features of reality are the laws of logic true? ; How do we know the truths of logic? ; and Could the laws of… …   Universalium Logic — For other uses, see Logic (disambiguation). Philosophy …   Wikipedia Modal algebra — In algebra and logic, a modal algebra is a structure such that is a Boolean algebra, is a unary operation on A satisfying and for all x, y in A. Modal algebras provide models of propos …   Wikipedia Modal verb — A modal verb (also modal, modal auxiliary verb, modal auxiliary) is a type of auxiliary verb that is used to indicate modality that is, likelihood, ability, permission, and obligation.[1]:p.33 The use of auxiliary verbs to express modality is… …   Wikipedia Deontic logic — is the field of logic that is concerned with obligation, permission, and related concepts. Alternatively, a deontic logic is a formal system that attempts to capture the essential logical features of these concepts. Typically, a deontic logic… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”