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# Modal logic

Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality. [1]

138 relations: Accessibility relation, Accident (philosophy), Accidental necessity, Alcubierre drive, Alethic modality, Alfred Tarski, Amir Pnueli, Analytic proof, Ancient Greek, Aristotle, Arthur Prior, Atom, Atomic number, Avicenna, Bertrand Russell, Bigfoot, Bjarni Jónsson, Blindsight, Boolean algebra, Boolean algebra (structure), Chrysippus, Clarence Irving Lewis, Classical modal logic, Closure operator, Computation tree logic, Computer science, Contraposition, Convention (norm), Cooper Harold Langford, Counterpart theory, Dana Scott, David Lewis (philosopher), De dicto and de re, De Interpretatione, De Morgan's laws, Deontic logic, Description logic, Diodorus Cronus, Doxastic logic, Duns Scotus, Dynamic logic (modal logic), Edward N. Zalta, Encyclopædia Britannica, Enthymeme, Epistemic modal logic, Equivalence relation, Essence, Euclidean relation, Evert Willem Beth, First-order logic, ... Expand index (88 more) »

## Accessibility relation

In modal logic, an accessibility relation R is a binary relation such that R \subseteq W \times W where W is a set of possible worlds.

## Accident (philosophy)

An accident, in philosophy, is an attribute that may or may not belong to a subject, without affecting its essence.

## Accidental necessity

In philosophy and logic, accidental necessity, often stated in its Latin form, necessitas per accidens, refers to the necessity attributed to the past by certain views of time.

## Alcubierre drive

The Alcubierre drive or Alcubierre warp drive (or Alcubierre metric, referring to metric tensor) is a speculative idea based on a solution of Einstein's field equations in general relativity as proposed by Mexican theoretical physicist Miguel Alcubierre, by which a spacecraft could achieve apparent faster-than-light travel if a configurable energy-density field lower than that of vacuum (that is, negative mass) could be created.

## Alethic modality

Alethic modality (from Greek ἀλήθεια.

## Alfred Tarski

Alfred Tarski (January 14, 1901 &ndash; October 26, 1983), born Alfred Teitelbaum,School of Mathematics and Statistics, University of St Andrews,, School of Mathematics and Statistics, University of St Andrews.

## Amir Pnueli

Amir Pnueli (אמיר פנואלי; April 22, 1941 – November 2, 2009) was an Israeli computer scientist and the 1996 Turing Award recipient.

## Analytic proof

In mathematics, an analytic proof is a proof of a theorem in analysis that only makes use of methods from analysis, and which does not predominantly make use of algebraic or geometrical methods.

## Ancient Greek

The Ancient Greek language includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD.

## Aristotle

Aristotle (Ἀριστοτέλης Aristotélēs,; 384–322 BC) was an ancient Greek philosopher and scientist born in the city of Stagira, Chalkidiki, in the north of Classical Greece.

## Arthur Prior

Arthur Norman Prior (4 December 1914 – 6 October 1969), usually cited as A. N. Prior, was a noted logician and philosopher.

## Atom

An atom is the smallest constituent unit of ordinary matter that has the properties of a chemical element.

## Atomic number

The atomic number or proton number (symbol Z) of a chemical element is the number of protons found in the nucleus of an atom.

## Avicenna

Avicenna (also Ibn Sīnā or Abu Ali Sina; ابن سینا; – June 1037) was a Persian polymath who is regarded as one of the most significant physicians, astronomers, thinkers and writers of the Islamic Golden Age.

## Bertrand Russell

Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 &ndash; 2 February 1970) was a British philosopher, logician, mathematician, historian, writer, social critic, political activist, and Nobel laureate.

## Bigfoot

In North American folklore, Bigfoot or Sasquatch is a hairy, upright-walking,ape-like being who reportedly dwells in the wilderness and leaves behind large footprints.

## Bjarni Jónsson

Bjarni Jónsson (February 15, 1920 – September 30, 2016) was an Icelandic mathematician and logician working in universal algebra, lattice theory, model theory and set theory.

## Blindsight

Blindsight is the ability of people who are cortically blind due to lesions in their striate cortex, also known as primary visual cortex or V1, to respond to visual stimuli that they do not consciously see.

## Boolean algebra

In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively.

## Boolean algebra (structure)

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.

## Chrysippus

Chrysippus of Soli (Χρύσιππος ὁ Σολεύς, Chrysippos ho Soleus) was a Greek Stoic philosopher.

## Clarence Irving Lewis

Clarence Irving Lewis (April 12, 1883 – February 3, 1964), usually cited as C. I. Lewis, was an American academic philosopher and the founder of conceptual pragmatism.

## 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 \Diamond A \equiv \lnot\Box\lnot A which is also closed under the rule A \equiv B \vdash \Box A\equiv\Box B. Alternatively one can give a dual definition of L by which L is classical iff it contains (as axiom or theorem) \Box A \equiv \lnot\Diamond\lnot A and is closed under the rule A \equiv B \vdash \Diamond A\equiv\Diamond B. The weakest classical system is sometimes referred to as E and is non-normal.

## Closure operator

In mathematics, a closure operator on a set S is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of S to itself which satisfies the following conditions for all sets X,Y\subseteq S |- | X \subseteq \operatorname(X) | (cl is extensive) |- | X\subseteq Y \Rightarrow \operatorname(X) \subseteq \operatorname(Y) | (cl is increasing) |- | \operatorname(\operatorname(X)).

## Computation tree logic

Computation tree logic (CTL) is a branching-time logic, meaning that its model of time is a tree-like structure in which the future is not determined; there are different paths in the future, any one of which might be an actual path that is realized.

## Computer science

Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations.

## Contraposition

In logic, contraposition is an inference that says that a conditional statement is logically equivalent to its contrapositive.

## Convention (norm)

A convention is a set of agreed, stipulated, or generally accepted standards, norms, social norms, or criteria, often taking the form of a custom.

## Cooper Harold Langford

Cooper Harold Langford (25 August 1895, Dublin, Logan County, Arkansas – 28 August 1964) was an analytic philosopher and mathematical logician who co-authored the book Symbolic Logic (1932) with C. I. Lewis.

## Counterpart theory

In philosophy, specifically in the area of modal metaphysics, counterpart theory is an alternative to standard (Kripkean) possible-worlds semantics for interpreting quantified modal logic.

## Dana Scott

Dana Stewart Scott (born October 11, 1932) is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California.

## David Lewis (philosopher)

David Kellogg Lewis (September 28, 1941 – October 14, 2001) was an American philosopher.

## De dicto and de re

De dicto and de re are two phrases used to mark a distinction in intentional statements, associated with the intentional operators in many such statements.

## De Interpretatione

De Interpretatione or On Interpretation (Greek: Περὶ Ἑρμηνείας, Peri Hermeneias) is the second text from Aristotle's Organon and is among the earliest surviving philosophical works in the Western tradition to deal with the relationship between language and logic in a comprehensive, explicit, and formal way.

## De Morgan's laws

In propositional logic and boolean algebra, De Morgan's laws are a pair of transformation rules that are both valid rules of inference.

## Deontic logic

Deontic logic is the field of philosophical logic that is concerned with obligation, permission, and related concepts.

## Description logic

Description logics (DL) are a family of formal knowledge representation languages.

## Diodorus Cronus

Diodorus Cronus (Διόδωρος Κρόνος; died c. 284 BCE) was a Greek philosopher and dialectician connected to the Megarian school.

## Doxastic logic

Doxastic logic is a type of logic concerned with reasoning about beliefs.

## Duns Scotus

John Duns, commonly called Duns Scotus (1266 – 8 November 1308), is generally considered to be one of the three most important philosopher-theologians of the High Middle Ages (together with Thomas Aquinas and William of Ockham).

## Dynamic logic (modal logic)

Dynamic logic is an extension of modal logic originally intended for reasoning about computer programs and later applied to more general complex behaviors arising in linguistics, philosophy, AI, and other fields.

## Edward N. Zalta

Edward N. Zalta (born March 16, 1952) is a senior research scholar at the Center for the Study of Language and Information.

## Encyclopædia Britannica

The Encyclopædia Britannica (Latin for "British Encyclopaedia"), published by Encyclopædia Britannica, Inc., is a general knowledge English-language encyclopaedia.

## Enthymeme

An enthymeme (ἐνθύμημα, enthumēma) is a rhetorical syllogism (a three-part deductive argument) used in oratorical practice.

## Epistemic modal logic

Epistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge.

## Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

## Essence

In philosophy, essence is the property or set of properties that make an entity or substance what it fundamentally is, and which it has by necessity, and without which it loses its identity.

## Euclidean relation

In mathematics, Euclidean relations are a class of binary relations that formalizes "Axiom 1" in Euclid's Elements: "Magnitudes which are equal to the same are equal to each other.".

## Evert Willem Beth

Evert Willem Beth (7 July 1908 &ndash; 12 April 1964) was a Dutch philosopher and logician, whose work principally concerned the foundations of mathematics.

## First-order logic

First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.

## Free On-line Dictionary of Computing

The Free On-line Dictionary of Computing (FOLDOC) is an online, searchable, encyclopedic dictionary of computing subjects.

## George Edward Hughes

George Edward Hughes (8 June 1918 – 4 March 1994) was an Irish-born New Zealand philosopher and logician whose principal scholarly works were concerned with modal logic and medieval philosophy.

The GNU Free Documentation License (GNU FDL or simply GFDL) is a copyleft license for free documentation, designed by the Free Software Foundation (FSF) for the GNU Project.

## Goldbach's conjecture

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics.

## Gottfried Wilhelm Leibniz

Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.

## Grammatical mood

In linguistics, grammatical mood (also mode) is a grammatical feature of verbs, used for signaling modality.

## Grammatical tense

In grammar, tense is a category that expresses time reference with reference to the moment of speaking.

## Harvard University

Harvard University is a private Ivy League research university in Cambridge, Massachusetts.

## Hennessy–Milner logic

In computer science, Hennessy–Milner logic (HML) is a dynamic logic used to specify properties of a labeled transition system (LTS), a structure similar to an automaton.

## Hybrid logic

Hybrid logic refers to a number of extensions to propositional modal logic with more expressive power, though still less than first-order logic.

## If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

## Indexicality

In semiotics, linguistics, anthropology and philosophy of language, indexicality is the phenomenon of a sign pointing to (or indexing) some object in the context in which it occurs.

## Interior algebra

In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set.

## Internet Encyclopedia of Philosophy

The Internet Encyclopedia of Philosophy (IEP) is a scholarly online encyclopedia, dealing with philosophy, philosophical topics, and philosophers.

## Interpretability logic

Interpretability logics comprise a family of modal logics that extend provability logic to describe interpretability or various related metamathematical properties and relations such as weak interpretability, Π1-conservativity, cointerpretability, tolerance, cotolerance, and arithmetic complexities.

## J.C.C. McKinsey

John Charles Chenoweth McKinsey (30 April 1908 – 26 October 1953) (also known as J. C. C. McKinsey or Chen McKinsey) was an American mathematician known for his work on mathematical logic and game theory., Stanford Historical Society He also made significant contributions to modal logic.

## James Garson

James Garson is an American philosopher and logician.

## Johan van Benthem (logician)

Johannes Franciscus Abraham Karel (Johan) van Benthem (born 12 June 1949 in Rijswijk) is a University Professor (universiteitshoogleraar) of logic at the University of Amsterdam at the Institute for Logic, Language and Computation and professor of philosophy at Stanford University (at CSLI).

## John Lemmon

Edward John Lemmon (1 June 1930 &ndash; 29 July 1966) was a logician and philosopher born in Sheffield, England.

## John McCarthy (computer scientist)

John McCarthy (September 4, 1927 – October 24, 2011) was an American computer scientist and cognitive scientist.

## Kripke semantics

Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Joyal.

## Latin

Latin (Latin: lingua latīna) is a classical language belonging to the Italic branch of the Indo-European languages.

## Linear temporal logic

In logic, linear temporal logic or linear-time temporal logic (LTL) is a modal temporal logic with modalities referring to time.

## Linguistic modality

In linguistics, modality is a feature of language that allows for communicating things about, or based on, situations which need not be actual.

## Logical equivalence

In logic, statements p and q are logically equivalent if they have the same logical content.

## Logical possibility

Logically possible refers to a proposition which can be the logical consequence of another, based on the axioms of a given system of logic.

## Many-valued logic

In logic, a many-valued logic (also multi- or multiple-valued logic) is a propositional calculus in which there are more than two truth values.

## Max Cresswell

Maxwell John Cresswell (born 19 November 1939, Wellington) is a New Zealand philosopher and logician, known for his work in modal logic.

## Max Planck Society

The Max Planck Society for the Advancement of Science (Max-Planck-Gesellschaft zur Förderung der Wissenschaften e. V.; abbreviated MPG) is a formally independent non-governmental and non-profit association of German research institutes founded in 1911 as the Kaiser Wilhelm Society and renamed the Max Planck Society in 1948 in honor of its former president, theoretical physicist Max Planck.

## Method of analytic tableaux

In proof theory, the semantic tableau (plural: tableaux, also called 'truth tree') is a decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic.

## Modal algebra

In algebra and logic, a modal algebra is a structure \langle A,\land,\lor,-,0,1,\Box\rangle such that.

## Modal fallacy

The formal fallacy of the modal fallacy is a special type of fallacy that occurs in modal logic.

## Modal operator

A modal connective (or modal operator) is a logical connective for modal logic.

## Modal realism

Modal realism is the view propounded by David Kellogg Lewis that all possible worlds are real in the same way as is the actual world: they are "of a kind with this world of ours." It is based on the following tenets: possible worlds exist; possible worlds are not different in kind from the actual world; possible worlds are irreducible entities; the term actual in actual world is indexical, i.e. any subject can declare their world to be the actual one, much as they label the place they are "here" and the time they are "now".

## Modal verb

A modal verb is a type of verb that is used to indicate modality – that is: likelihood, ability, permission and obligation, and advice.

## Naturalistic fallacy

In philosophical ethics, the term "naturalistic fallacy" was introduced by British philosopher G. E. Moore in his 1903 book Principia Ethica.

## Negation

In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P (¬P), which is interpreted intuitively as being true when P is false, and false when P is true.

## Nicholas Rescher

Nicholas Rescher (born 15 July 1928) is a German-American philosopher at the University of Pittsburgh.

## Norm (philosophy)

Norms are concepts (sentences) of practical import, oriented to effecting an action, rather than conceptual abstractions that describe, explain, and express.

## Normal modal logic

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

## Obligation

An obligation is a course of action that someone is required to take, whether legal or moral.

## Philo the Dialectician

Philo the Dialectician (Φίλων; fl. 300 BC) was a dialectic philosopher of the Megarian school.

## Philosopher

A philosopher is someone who practices philosophy, which involves rational inquiry into areas that are outside either theology or science.

## Physical law

A physical law or scientific law is a theoretical statement "inferred from particular facts, applicable to a defined group or class of phenomena, and expressible by the statement that a particular phenomenon always occurs if certain conditions be present." Physical laws are typically conclusions based on repeated scientific experiments and observations over many years and which have become accepted universally within the scientific community.

## Physical Review

Physical Review is an American peer-reviewed scientific journal established in 1893 by Edward Nichols.

## Physicalism

In philosophy, physicalism is the ontological thesis that "everything is physical", that there is "nothing over and above" the physical, or that everything supervenes on the physical.

## Polish notation

Polish notation (PN), also known as normal Polish notation (NPN), Łukasiewicz notation, Warsaw notation, Polish prefix notation or simply prefix notation, is a mathematical notation in which operators precede their operands, in contrast to reverse Polish notation (RPN) in which operators follow their operands.

## Possible world

In philosophy and logic, the concept of a possible world is used to express modal claims.

## Potentiality and actuality

In philosophy, potentiality and actuality are principles of a dichotomy which Aristotle used to analyze motion, causality, ethics, and physiology in his Physics, Metaphysics, Nicomachean Ethics and De Anima, which is about the human psyche.

## Preorder

In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive.

## Principle of bivalence

In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false.

## Prior Analytics

The Prior Analytics (Ἀναλυτικὰ Πρότερα; Analytica Priora) is Aristotle's work on deductive reasoning, which is known as his syllogistic.

## Problem of future contingents

Future contingent propositions (or simply, future contingents) are statements about states of affairs in the future that are contingent: neither necessarily true nor necessarily false.

## Propositional calculus

Propositional calculus is a branch of logic.

## Propositional function

A propositional function in logic, is a sentence expressed in a way that would assume the value of true or false, except that within the sentence is a variable (x) that is not defined or specified, which leaves the statement undetermined.

## Provability logic

Provability logic is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'.

## Reflexive relation

In mathematics, a binary relation R over a set X is reflexive if every element of X is related to itself.

## Regular modal logic

In modal logic, a regular modal logic L is a modal logic closed under the duality of the modal operators: \Diamond A \equiv \lnot\Box\lnot A and the rule (A\land B)\to C \vdash (\Box A\land\Box B)\to\Box C. Every regular modal logic is classical, and every normal modal logic is regular and hence classical.

## Relevance logic

Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related.

## Rhetoric

Rhetoric is the art of discourse, wherein a writer or speaker strives to inform, persuade, or motivate particular audiences in specific situations.

## Robert Goldblatt

Robert Ian Goldblatt (born 1949) is a mathematical logician at the School of Mathematics and Statistics at Victoria University, Wellington, New Zealand, and a member of the Centre for Logic, Language and Computation.

Robert Merrihew Adams (born September 8, 1937), known to intimates as "Bob", is an American analytic philosopher of metaphysics, religion and morality.

## Ruth Barcan Marcus

Ruth Barcan Marcus (born Ruth C. Barcan; August 2, 1921 &ndash; February 19, 2012) was an American philosopher and logician who developed the Barcan formula.

## Saul Kripke

Saul Aaron Kripke (born November 13, 1940) is an American philosopher and logician.

## Scholasticism

Scholasticism is a method of critical thought which dominated teaching by the academics ("scholastics", or "schoolmen") of medieval universities in Europe from about 1100 to 1700, and a program of employing that method in articulating and defending dogma in an increasingly pluralistic context.

## Semantics

Semantics (from σημαντικός sēmantikós, "significant") is the linguistic and philosophical study of meaning, in language, programming languages, formal logics, and semiotics.

## Serial relation

In set theory, a serial relation is a binary relation R for which every element of the domain has a corresponding range element (∀ x ∃ y x R y).

## Socrates

Socrates (Sōkrátēs,; – 399 BC) was a classical Greek (Athenian) philosopher credited as one of the founders of Western philosophy, and as being the first moral philosopher, of the Western ethical tradition of thought.

## Speed of light

The speed of light in vacuum, commonly denoted, is a universal physical constant important in many areas of physics.

## Stanford Encyclopedia of Philosophy

The Stanford Encyclopedia of Philosophy (SEP) combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users.

## Strict conditional

In logic, a strict conditional is a conditional governed by a modal operator, that is, a logical connective of modal logic.

## Structural proof theory

In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof.

## Subliminal stimuli

Subliminal stimuli (the prefix sup- literally "below, or less than", while the prefix sub- literally "up to"), contrary to supraliminal stimuli or "above threshold", are any sensory stimuli below an individual's threshold for conscious perception.

## Symmetric relation

In mathematics and other areas, a binary relation R over a set X is symmetric if it holds for all a and b in X that a is related to b if and only if b is related to a. In mathematical notation, this is: Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.

## Temporal logic

In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time.

## Theophrastus

Theophrastus (Θεόφραστος Theόphrastos; c. 371 – c. 287 BC), a Greek native of Eresos in Lesbos,Gavin Hardy and Laurence Totelin, Ancient Botany, 2015, p. 8.

## Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms.

## Time

Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past through the present to the future.

## Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

## Transitive relation

In mathematics, a binary relation over a set is transitive if whenever an element is related to an element and is related to an element then is also related to.

## Truth value

In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth.

## Two-dimensionalism

Two-dimensionalism is an approach to semantics in analytic philosophy.

## Unary operation

In mathematics, a unary operation is an operation with only one operand, i.e. a single input.

## Vacuous truth

In mathematics and logic, a vacuous truth is a statement that asserts that all members of the empty set have a certain property.

## Vaughan Pratt

Vaughan Pratt (born on April 12, 1944) is a Professor Emeritus at Stanford University, who was an early pioneer in the field of computer science.

## Willard Van Orman Quine

Willard Van Orman Quine (known to intimates as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century." From 1930 until his death 70 years later, Quine was continually affiliated with Harvard University in one way or another, first as a student, then as a professor of philosophy and a teacher of logic and set theory, and finally as a professor emeritus who published or revised several books in retirement.

## William of Ockham

William of Ockham (also Occam, from Gulielmus Occamus; 1287 – 1347) was an English Franciscan friar and scholastic philosopher and theologian, who is believed to have been born in Ockham, a small village in Surrey.

## Wormhole

A wormhole is a concept that represents a solution of the Einstein field equations: a non-trivial resolution of the Ehrenfest paradox structure linking separate points in spacetime.

## References

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