modal logic

﻿
modal logic
Formal systems incorporating modalities such as necessity, possibility, impossibility, contingency, strict implication, and certain other closely related concepts.

The most straightforward way of constructing a modal logic is to add to some standard nonmodal logical system a new primitive operator intended to represent one of the modalities, to define other modal operators in terms of it, and to add axioms and/or transformation rules involving those modal operators. For example, one may add the symbol L, which means "It is necessary that," to classical propositional calculus; thus, Lp is read as "It is necessary that p." The possibility operator M ("It is possible that") may be defined in terms of L as Mp = ¬L¬p (where ¬ means "not"). In addition to the axioms and rules of inference of classical propositional logic, such a system might have two axioms and one rule of inference of its own. Some characteristic axioms of modal logic are: (A1) Lp ⊃ p and (A2) L(p ⊃ q) ⊃ (Lp ⊃ Lq). The new rule of inference in this system is the Rule of Necessitation: If p is a theorem of the system, then so is Lp. Stronger systems of modal logic can be obtained by adding additional axioms. Some add the axiom Lp ⊃ LLp; others add the axiom Mp ⊃ LMp.

* * *

branch of logic that deals with modalities (such properties of propositions as necessity, contingency, possibility, and impossibility), as opposed to truth and falsity; thus the statements “Some men may be immortal” and “Men are necessarily social animals” are modal propositions. Although modal syllogisms were considered by Aristotle, modal logic remains today an uncertain field. Modern attempts to deal with the problem are found in the many-valued logics, which allow other truth-values between truth and falsity, and in systems of strict implication—systems of theorems that differ somewhat depending upon the relations between the different modalities that are set forth in their axioms. Compare truth-value.

* * *

Universalium. 2010.

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

• modal logic — mo dal log ic, n. A system of logic which studies how to combine propositions which include the concepts of necessity, possibility, and obligation. [PJC] …   The Collaborative International Dictionary of English

• modal logic — A logic studying the notions of necessity and possibility. Modal logic was of great importance historically, particularly in the light of various doctrines concerning the necessary properties of the deity, but was not a central topic of modern… …   Philosophy dictionary

• modal logic — noun 1. the logical study of necessity and possibility • Hypernyms: ↑logic 2. a system of logic whose formal properties resemble certain moral and epistemological concepts • Hypernyms: ↑symbolic logic, ↑mathematical logic, ↑formal logic • …   Useful english dictionary

• modal logic — noun Any formal system that attempts to deal with modalities, such as possibility and necessity, but also obligation and permission. See Also: deontic logic, doxastic logic, epistemic logic …   Wiktionary

• Epistemic modal logic — is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition dating back to Ancient Greece, epistemic logic is a much more recent development with applications in many fields …   Wikipedia

• Dynamic logic (modal logic) — For the subject in digital electronics also known as clocked logic, see dynamic logic (digital electronics). Dynamic logic is an extension of modal logic originally intended for reasoning about computer programs and later applied to more general… …   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