Modal logic is a collection of
formal system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A form ...
s developed to represent statements about
necessity and possibility. It plays a major role in
philosophy of language
In analytic philosophy, philosophy of language investigates the nature of language and the relations between language, language users, and the world. Investigations may include inquiry into the nature of meaning, intentionality, reference, ...
,
epistemology
Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics.
Episte ...
,
metaphysics
Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
, and
natural language semantics
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...
. Modal logics extend other systems by adding
unary operators
and
, representing possibility and necessity respectively. For instance the modal formula
can be read as "possibly
" while
can be read as "necessarily
". Modal logics can be used to represent different phenomena depending on what kind of necessity and possibility is under consideration. When
is used to represent
epistemic necessity,
states that
is epistemically necessary, or in other words that it is known. When
is used to represent
deontic necessity,
states that
is a moral or legal obligation.
In the standard
relational semantics for modal logic, formulas are assigned truth values relative to a ''
possible world
A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional logic, intensional and mod ...
''. A formula's truth value at one possible world can depend on the truth values of other formulas at other
''accessible'' possible worlds Possible Worlds may refer to:
* Possible worlds, concept in philosophy
* ''Possible Worlds'' (play), 1990 play by John Mighton
** ''Possible Worlds'' (film), 2000 film by Robert Lepage, based on the play
* Possible Worlds (studio)
* ''Possible Wo ...
. In particular,
is true at a world if
is true at ''some'' accessible possible world, while
is true at a world if
is true at ''every'' accessible possible world. A variety of proof systems exist which are sound and complete with respect to the semantics one gets by restricting the accessibility relation. For instance, the deontic modal logic D is sound and complete if one requires the accessibility relation to be
serial.
While the intuition behind modal logic dates back to antiquity, the first modal
axiomatic system
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contai ...
s were developed by
C. I. Lewis
Clarence Irving Lewis (April 12, 1883 – February 3, 1964), usually cited as C. I. Lewis, was an American academic philosopher. He is considered the progenitor of modern modal logic and the founder of conceptual pragmatism. First a noted logic ...
in 1912. The now-standard relational semantics emerged in the mid twentieth century from work by
Arthur Prior
Arthur Norman Prior (4 December 1914 – 6 October 1969), usually cited as A. N. Prior, was a New Zealand–born logician and philosopher. Prior (1957) founded tense logic, now also known as temporal logic, and made important contribution ...
,
Jaakko Hintikka
Kaarlo Jaakko Juhani Hintikka (12 January 1929 – 12 August 2015) was a Finnish philosopher and logician.
Life and career
Hintikka was born in Helsingin maalaiskunta (now Vantaa).
In 1953, he received his doctorate from the University of Helsin ...
, and
Saul Kripke
Saul Aaron Kripke (; November 13, 1940 – September 15, 2022) was an American philosopher and logician in the analytic tradition. He was a Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emerit ...
. Recent developments include alternative
topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
semantics such as
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
Kripke sem ...
as well as applications of the relational semantics beyond its original philosophical motivation.
Such applications include
game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
,
moral
A moral (from Latin ''morālis'') is a message that is conveyed or a lesson to be learned from a story or event. The moral may be left to the hearer, reader, or viewer to determine for themselves, or may be explicitly encapsulated in a maxim. A ...
and
legal theory
Jurisprudence, or legal theory, is the theoretical study of the propriety of law. Scholars of jurisprudence seek to explain the nature of law in its most general form and they also seek to achieve a deeper understanding of legal reasoning a ...
,
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 an ...
,
multiverse-based set theory, and
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 o ...
.
Syntax of modal operators
Modal logic differs from other kinds of logic in that it uses modal
operators such as
and
. The former is conventionally read aloud as "necessarily", and can be used to represent notions such as moral or legal
obligation
An obligation is a course of action that someone is required to take, whether legal or moral. Obligations are constraints; they limit freedom. People who are under obligations may choose to freely act under obligations. Obligation exists when the ...
,
knowledge
Knowledge can be defined as awareness of facts or as practical skills, and may also refer to familiarity with objects or situations. Knowledge of facts, also called propositional knowledge, is often defined as true belief that is distinc ...
,
historical inevitability, among others. The latter is typically read as "possibly" and can be used to represent notions including
permission,
ability
Abilities are powers an agent has to perform various actions. They include common abilities, like walking, and rare abilities, like performing a double backflip. Abilities are intelligent powers: they are guided by the person's intention and exec ...
, compatibility with
evidence
Evidence for a proposition is what supports this proposition. It is usually understood as an indication that the supported proposition is true. What role evidence plays and how it is conceived varies from field to field.
In epistemology, evidenc ...
. While
well formed formula
In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can b ...
s of modal logic include non-modal formulas such as
, it also contains modal ones such as
,
,
, and so on.
Thus, the
language
Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...
of basic
propositional logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
can be
defined recursively as follows.
#If
is an atomic formula, then
is a formula of
.
#If
is a formula of
, then
is too.
#If
and
are formulas of
, then
is too.
#If
is a formula of
, then
is too.
#If
is a formula of
, then
is too.
Modal operators can be added to other kinds of logic by introducing rules analogous to #4 and #5 above. Modal
predicate logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
is one widely used variant which includes formulas such as
. In systems of modal logic where
and
are
duals
''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers.
Track listing
:* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, Pas ...
,
can be taken as an abbreviation for
, thus eliminating the need for a separate syntactic rule to introduce it. However, separate syntactic rules are necessary in systems where the two operators are not interdefinable.
Common notational variants include symbols such as