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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. Epis ...
,
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 \Diamond and \Box, representing possibility and necessity respectively. For instance the modal formula \Diamond P can be read as "possibly P" while \Box P can be read as "necessarily P". Modal logics can be used to represent different phenomena depending on what kind of necessity and possibility is under consideration. When \Box is used to represent epistemic necessity, \Box P states that P is epistemically necessary, or in other words that it is known. When \Box is used to represent deontic necessity, \Box P states that P 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 and modal logic. Their me ...
''. 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, \Diamond P is true at a world if P is true at ''some'' accessible possible world, while \Box P is true at a world if P 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 contains ...
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, Jaakko Hintikka, 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 em ...
. 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 ...
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, moral 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 ...
,
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 a ...
, 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 \Box and \Diamond. 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 Descriptive knowledge, awareness of facts or as Procedural knowledge, practical skills, and may also refer to Knowledge by acquaintance, familiarity with objects or situations. Knowledge of facts, also called pro ...
, 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. While well formed formulas of modal logic include non-modal formulas such as P \land Q, it also contains modal ones such as \Box(P \land Q), P \land \Box Q, \Box(\Diamond P \land \Diamond Q), 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 ...
\mathcal 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 \phi is an atomic formula, then \phi is a formula of \mathcal. #If \phi is a formula of \mathcal, then \neg \phi is too. #If \phi and \psi are formulas of \mathcal, then \phi \land \psi is too. #If \phi is a formula of \mathcal, then \Diamond \phi is too. #If \phi is a formula of \mathcal, then \Box \phi 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 \forall x \Diamond P(x) . In systems of modal logic where \Box and \Diamond 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, P ...
, \Box \phi can be taken as an abbreviation for \neg \Diamond \neg \phi, 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 /math> and \langle K \rangle in systems of modal logic used to represent knowledge and /math> and \langle B \rangle in those used to represent belief. These notations are particularly common in systems which use multiple modal operators simultaneously. For instance, a combined epistemic-deontic logic could use the formula langle D \rangle P read as "I know P is permitted". Systems of modal logic can include infinitely many modal operators distinguished by indices, i.e. \Box_1, \Box_2, \Box_3, and so on.


Semantics


Relational semantics


Basic notions

The standard semantics for modal logic is called the ''relational semantics''. In this approach, the truth of a formula is determined relative to a point which is often called 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 and modal logic. Their me ...
''. For a formula that contains a modal operator, its truth value can depend on what is true at other
accessible Accessibility is the design of products, devices, services, vehicles, or environments so as to be usable by people with disabilities. The concept of accessible design and practice of accessible development ensures both "direct access" (i. ...
worlds. Thus, the relational semantics interprets formulas of modal logic using models defined as follows. * A ''relational model'' is a tuple \mathfrak = \langle W, R, V \rangle where: # W is a set of possible worlds # R is a binary relation on W # V is a valuation function which assigns a truth value to each pair of an atomic formula and a world, (i.e. V: W \times F \to \ where F is the set of atomic formulae) The set W is often called the ''universe''. The binary relation R is called an
accessibility relation An accessibility relation is a relation which plays a key role in assigning truth values to sentences in the relational semantics for modal logic. In relational semantics, a modal formula's truth value at a '' possible world'' w can depend on ...
, and it controls which worlds can "see" each other for the sake of determining what is true. For example, w R u means that the world u is accessible from world w. That is to say, the state of affairs known as u is a live possibility for w. Finally, the function V is known as a
valuation function Valuation may refer to: Economics *Valuation (finance), the determination of the economic value of an asset or liability **Real estate appraisal, sometimes called ''property valuation'' (especially in British English), the appraisal of land or bui ...
. It determines which
atomic formula In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformu ...
s are true at which worlds. Then we recursively define the truth of a formula at a world w in a model \mathfrak: * \mathfrak, w \models P iff V(w, P)=1 * \mathfrak, w \models \neg P iff w \not \models P * \mathfrak, w \models (P \wedge Q) iff w \models P and w \models Q * \mathfrak, w \models \Box P iff for every element u of W, if w R u then u \models P * \mathfrak, w \models \Diamond P iff for some element u of W, it holds that w R u and u \models P According to this semantics, a formula is ''necessary'' with respect to a world w if it holds at every world that is accessible from w. It is ''possible'' if it holds at some world that is accessible from w. Possibility thereby depends upon the accessibility relation R, which allows us to express the relative nature of possibility. For example, we might say that given our laws of physics it is not possible for humans to travel faster than the speed of light, but that given other circumstances it could have been possible to do so. Using the accessibility relation we can translate this scenario as follows: At all of the worlds accessible to our own world, it is not the case that humans can travel faster than the speed of light, but at one of these accessible worlds there is ''another'' world accessible from ''those'' worlds but not accessible from our own at which humans can travel faster than the speed of light.


Frames and completeness

The choice of accessibility relation alone can sometimes be sufficient to guarantee the truth or falsity of a formula. For instance, consider a model \mathfrak whose accessibility relation is reflexive. Because the relation is reflexive, we will have that \mathfrak,w \models P \rightarrow \Diamond P for any w \in G regardless of which valuation function is used. For this reason, modal logicians sometimes talk about ''frames'', which are the portion of a relational model excluding the valuation function. * A ''relational frame'' is a pair \mathfrak = \langle G, R \rangle where G is a set of possible worlds, R is a binary relation on G. The different systems of modal logic are defined using ''frame conditions''. A frame is called: * reflexive if ''w R w'', for every ''w'' in ''G'' *
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
if ''w R u'' implies ''u R w'', for all ''w'' and ''u'' in ''G'' * transitive if ''w R u'' and ''u R q'' together imply ''w R q'', for all ''w'', ''u'', ''q'' in ''G''. * serial if, for every ''w'' in ''G'' there is some ''u'' in ''G'' such that ''w R u''. * Euclidean if, for every ''u'', ''t'', and ''w'', ''w R u'' and ''w R t'' implies ''u R t'' (by symmetry, it also implies ''t R u'', as well as ''t R t'' and ''u R u'') The logics that stem from these frame conditions are: *''K'' := no conditions *''D'' := serial *''T'' := reflexive *''B'' := reflexive and symmetric *''S4'' := reflexive and transitive *''S5'' := reflexive and Euclidean The Euclidean property along with reflexivity yields symmetry and transitivity. (The Euclidean property can be obtained, as well, from symmetry and transitivity.) Hence if the accessibility relation ''R'' is reflexive and Euclidean, ''R'' is provably
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
and transitive as well. Hence for models of S5, ''R'' is an equivalence relation, because ''R'' is reflexive, symmetric and transitive. We can prove that these frames produce the same set of valid sentences as do the frames where all worlds can see all other worlds of ''W'' (''i.e.'', where ''R'' is a "total" relation). This gives the corresponding ''modal graph'' which is total complete (''i.e.'', no more edges (relations) can be added). For example, in any modal logic based on frame conditions: : w \models \Diamond P if and only if for some element ''u'' of ''G'', it holds that u \models P and ''w R u''. If we consider frames based on the total relation we can just say that : w \models \Diamond P if and only if for some element ''u'' of ''G'', it holds that u \models P. We can drop the accessibility clause from the latter stipulation because in such total frames it is trivially true of all ''w'' and ''u'' that ''w R u''. But note that this does not have to be the case in all S5 frames, which can still consist of multiple parts that are fully connected among themselves but still disconnected from each other. All of these logical systems can also be defined axiomatically, as is shown in the next section. For example, in S5, the axioms P \implies \Box\Diamond P, \Box P \implies \Box\Box P and \Box P \implies P (corresponding to ''symmetry'', ''transitivity'' and ''reflexivity'', respectively) hold, whereas at least one of these axioms does not hold in each of the other, weaker logics.


Topological semantics

Modal logic has also been interpreted using topological structures. For instance, the ''Interior Semantics'' interprets formulas of modal logic as follows. A ''topological model'' is a tuple \Chi = \langle X, \tau, V \rangle where \langle X, \tau \rangle is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
and V is a valuation function which maps each atomic formula to some subset of X. The basic interior semantics interprets formulas of modal logic as follows: * \Chi, x \models P iff x \in V(P) * \Chi, x \models \neg \phi iff \Chi, x \not\models \phi * \Chi, x \models \phi \land \chi iff \Chi, x \models \phi and \Chi, x \models \chi * \Chi, x \models \Box \phi iff for some U \in \tau we have both that x \in U and also that \Chi, y \models \phi for all y \in U Topological approaches subsume relational ones, allowing non-normal modal logics. The extra structure they provide also allows a transparent way of modeling certain concepts such as the evidence or justification one has for one's beliefs. Topological semantics is widely used in recent work in formal epistemology and has antecedents in earlier work such as David Lewis and Angelika Kratzer's logics for counterfactuals.


Axiomatic systems

The first formalizations of modal logic were
axiomatic An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
. Numerous variations with very different properties have been proposed since
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 ...
began working in the area in 1912. Hughes and Cresswell (1996), for example, describe 42
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
and 25 non-normal modal logics. Zeman (1973) describes some systems Hughes and Cresswell omit. Modern treatments of modal logic begin by augmenting the
propositional calculus 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 ...
with two unary operations, one denoting "necessity" and the other "possibility". The notation of
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 ...
, much employed since, denotes "necessarily ''p''" by a prefixed "box" (□''p'') whose scope is established by parentheses. Likewise, a prefixed "diamond" (◇''p'') denotes "possibly ''p''". Similar to the quantifiers in
first-order 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 ...
, "necessarily ''p''" (□''p'') does not assume the range of quantification (the set of accessible possible worlds in
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 Jo ...
) to be non-empty, whereas "possibly ''p''" (◇''p'') often implicitly assumes \Diamond\top (viz. the set of accessible possible worlds is non-empty). Regardless of notation, each of these operators is definable in terms of the other in classical modal logic: * □''p'' (necessarily ''p'') is equivalent to ("not possible that not-''p''") * ◇''p'' (possibly ''p'') is equivalent to ("not necessarily not-''p''") Hence □ and ◇ form a
dual pair In mathematics, a dual system, dual pair, or duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces X and Y over \mathbb and a non-degenerate bilinear map b : X \times Y \to \mathbb. Duality theory, the study of dual ...
of operators. In many modal logics, the necessity and possibility operators satisfy the following analogues of
de Morgan's laws In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British math ...
from
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
: :"It is not necessary that ''X''" is
logically equivalent Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
to "It is possible that not ''X''". :"It is not possible that ''X''" is logically equivalent to "It is necessary that not ''X''". Precisely what axioms and rules must be added to the
propositional calculus 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 ...
to create a usable system of modal logic is a matter of philosophical opinion, often driven by the theorems one wishes to prove; or, in computer science, it is a matter of what sort of computational or deductive system one wishes to model. Many modal logics, known collectively as
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 po ...
s, include the following rule and axiom: * N, Necessitation Rule: If ''p'' is a
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
/ tautology (of any system/model invoking N), then □''p'' is likewise a theorem (i.e. (\models p) \implies (\models \Box p) ). * K, Distribution Axiom: The weakest
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 po ...
, named "''K''" in honor of
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 em ...
, is simply the
propositional calculus 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 ...
augmented by □, the rule N, and the axiom K. ''K'' is weak in that it fails to determine whether a proposition can be necessary but only contingently necessary. That is, it is not a theorem of ''K'' that if □''p'' is true then □□''p'' is true, i.e., that necessary truths are "necessarily necessary". If such perplexities are deemed forced and artificial, this defect of ''K'' is not a great one. In any case, different answers to such questions yield different systems of modal logic. Adding axioms to ''K'' gives rise to other well-known modal systems. One cannot prove in ''K'' that if "''p'' is necessary" then ''p'' is true. The axiom T remedies this defect: *T, Reflexivity Axiom: (If ''p'' is necessary, then ''p'' is the case.) T holds in most but not all modal logics. Zeman (1973) describes a few exceptions, such as ''S10''. Other well-known elementary axioms are: *4: \Box p \to \Box \Box p *B: p \to \Box \Diamond p *D: \Box p \to \Diamond p *5: \Diamond p \to \Box \Diamond p These yield the systems (axioms in bold, systems in italics): *''K'' := K + N *''T'' := ''K'' + T *''S4'' := ''T'' + 4 *''S5'' := ''T'' + 5 *''D'' := ''K'' + D. ''K'' through ''S5'' form a nested hierarchy of systems, making up the core of
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 po ...
. But specific rules or sets of rules may be appropriate for specific systems. For example, in
deontic logic Deontic logic is the field of philosophical 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. It ...
, \Box p \to \Diamond p (If it ought to be that ''p'', then it is permitted that ''p'') seems appropriate, but we should probably not include that p \to \Box \Diamond p. In fact, to do so is to commit the
naturalistic fallacy In philosophical ethics, the naturalistic fallacy is the claim that any reductive explanation of good, in terms of natural properties such as ''pleasant'' or ''desirable'', is false. The term was introduced by British philosopher G. E. Moore in ...
(i.e. to state that what is natural is also good, by saying that if ''p'' is the case, ''p'' ought to be permitted). The commonly employed system ''S5'' simply makes all modal truths necessary. For example, if ''p'' is possible, then it is "necessary" that ''p'' is possible. Also, if ''p'' is necessary, then it is necessary that ''p'' is necessary. Other systems of modal logic have been formulated, in part because ''S5'' does not describe every kind of modality of interest.


Structural proof theory

Sequent calculi and systems of natural deduction have been developed for several modal logics, but it has proven hard to combine generality with other features expected of good structural proof theories, such as purity (the proof theory does not introduce extra-logical notions such as labels) and analyticity (the logical rules support a clean notion of analytic proof). More complex calculi have been applied to modal logic to achieve generality.


Decision methods

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. An analytic tableau is a tree structure compu ...
provide the most popular decision method for modal logics.


Modal logics in philosophy


Alethic logic

Modalities of necessity and possibility are called ''alethic'' modalities. They are also sometimes called ''special'' modalities, from the
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
''species''. Modal logic was first developed to deal with these concepts, and only afterward was extended to others. For this reason, or perhaps for their familiarity and simplicity, necessity and possibility are often casually treated as ''the'' subject matter of modal logic. Moreover, it is easier to make sense of relativizing necessity, e.g. to legal, physical,
nomological In philosophy, nomology refers to a "science of laws" based on the theory that it is possible to elaborate descriptions dedicated not to particular aspects of reality but inspired by a scientific vision of universal validity expressed by scientific ...
, epistemic, and so on, than it is to make sense of relativizing other notions. In
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 \leftrightarrow \lnot\Box\lnot A that is also closed under the rule \frac. Alternatively, one can gi ...
, a proposition is said to be *possible if it is ''not necessarily false'' (regardless of whether it is actually true or actually false); *necessary if it is ''not possibly false'' (i.e. true and necessarily true); *contingent if it is ''not necessarily false'' and ''not necessarily true'' (i.e. possible but not necessarily true); *impossible if it is ''not possibly true'' (i.e. false and necessarily false). In classical modal logic, therefore, the notion of either possibility or necessity may be taken to be basic, where these other notions are defined in terms of it in the manner of
De Morgan duality In propositional calculus, propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both Validity (logic), valid rule of inference, rules of inference. They are na ...
. Intuitionistic modal logic treats possibility and necessity as not perfectly symmetric. For example, suppose that while walking to the convenience store we pass Friedrich's house, and observe that the lights are off. On the way back, we observe that they have been turned on. * "Somebody or something turned the lights on" is ''necessary''. * "Friedrich turned the lights on", "Friedrich's roommate Max turned the lights on" and "A burglar named Adolf broke into Friedrich's house and turned the lights on" are ''contingent''. * All of the above statements are ''possible''. * It is ''impossible'' that
Socrates Socrates (; ; –399 BC) was a Greek philosopher from Athens who is credited as the founder of Western philosophy and among the first moral philosophers of the ethical tradition of thought. An enigmatic figure, Socrates authored no te ...
(who has been dead for over two thousand years) turned the lights on. (Of course, this analogy does not apply alethic modality in a ''truly'' rigorous fashion; for it to do so, it would have to axiomatically make such statements as "human beings cannot rise from the dead", "Socrates was a human being and not an immortal vampire", and "we did not take hallucinogenic drugs which caused us to falsely believe the lights were on", ''ad infinitum''. Absolute certainty of truth or falsehood exists only in the sense of logically constructed abstract concepts such as "it is impossible to draw a triangle with four sides" and "all bachelors are unmarried".) For those having difficulty with the concept of something being possible but not true, the meaning of these terms may be made more comprehensible by thinking of multiple "possible worlds" (in the sense of
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
) or "alternate universes"; something "necessary" is true in all possible worlds, something "possible" is true in at least one possible world. These "possible world semantics" are formalized with
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 Jo ...
.


Physical possibility

Something is physically, or nomically, possible if it is permitted by the
laws of physics Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) ...
. For example, current theory is thought to allow for there to be an
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, ...
with an
atomic number The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of every ...
of 126, even if there are no such atoms in existence. In contrast, while it is logically possible to accelerate beyond the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, See also Feinberg's later paper: Phys. Rev. D 17, 1651 (1978) modern science stipulates that it is not physically possible for material particles or information.


Metaphysical possibility

Philosophers A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...
debate if objects have properties independent of those dictated by scientific laws. For example, it might be metaphysically necessary, as some who advocate physicalism have thought, that all thinking beings have bodies and can experience the passage of
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, ...
.
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 em ...
has argued that every person necessarily has the parents they do have: anyone with different parents would not be the same person.
Metaphysical possibility Subjunctive possibility (also called alethic possibility) is a form of modality studied in modal logic. Subjunctive possibilities are the sorts of possibilities considered when conceiving counterfactual situations; subjunctive modalities are modal ...
has been thought to be more restricting than bare logical possibility (i.e., fewer things are metaphysically possible than are logically possible). However, its exact relation (if any) to logical possibility or to physical possibility is a matter of dispute. Philosophers also disagree over whether metaphysical truths are necessary merely "by definition", or whether they reflect some underlying deep facts about the world, or something else entirely.


Epistemic logic

Epistemic modalities (from the Greek ''episteme'', knowledge), deal with the ''certainty'' of sentences. The □ operator is translated as "x knows that…", and the ◇ operator is translated as "For all x knows, it may be true that…" In ordinary speech both metaphysical and epistemic modalities are often expressed in similar words; the following contrasts may help: A person, Jones, might reasonably say ''both'': (1) "No, it is ''not'' possible that Bigfoot exists; I am quite certain of that"; ''and'', (2) "Sure, it's ''possible'' that Bigfoots could exist". What Jones means by (1) is that, given all the available information, there is no question remaining as to whether Bigfoot exists. This is an epistemic claim. By (2) he makes the ''metaphysical'' claim that it is ''possible for'' Bigfoot to exist, ''even though he does not'': there is no physical or biological reason that large, featherless, bipedal creatures with thick hair could not exist in the forests of North America (regardless of whether or not they do). Similarly, "it is possible for the person reading this sentence to be fourteen feet tall and named Chad" is ''metaphysically'' true (such a person would not somehow be prevented from doing so on account of their height and name), but not ''alethically'' true unless you match that description, and not ''epistemically'' true if it's known that fourteen-foot-tall human beings have never existed. From the other direction, Jones might say, (3) "It is ''possible'' that Goldbach's conjecture is true; but also ''possible'' that it is false", and ''also'' (4) "if it ''is'' true, then it is necessarily true, and not possibly false". Here Jones means that it is ''epistemically possible'' that it is true or false, for all he knows (Goldbach's conjecture has not been proven either true or false), but if there ''is'' a proof (heretofore undiscovered), then it would show that it is not ''logically'' possible for Goldbach's conjecture to be false—there could be no set of numbers that violated it. Logical possibility is a form of ''alethic'' possibility; (4) makes a claim about whether it is possible (i.e., logically speaking) that a mathematical truth to have been false, but (3) only makes a claim about whether it is possible, for all Jones knows, (i.e., speaking of certitude) that the mathematical claim is specifically either true or false, and so again Jones does not contradict himself. It is worthwhile to observe that Jones is not necessarily correct: It is possible (epistemically) that Goldbach's conjecture is both true and unprovable. Epistemic possibilities also bear on the actual world in a way that metaphysical possibilities do not. Metaphysical possibilities bear on ways the world ''might have been,'' but epistemic possibilities bear on the way the world ''may be'' (for all we know). Suppose, for example, that I want to know whether or not to take an umbrella before I leave. If you tell me "it is ''possible that'' it is raining outside" – in the sense of epistemic possibility – then that would weigh on whether or not I take the umbrella. But if you just tell me that "it is ''possible for'' it to rain outside" – in the sense of ''metaphysical possibility'' – then I am no better off for this bit of modal enlightenment. Some features of epistemic modal logic are in debate. For example, if ''x'' knows that ''p'', does ''x'' know that it knows that ''p''? That is to say, should □''P'' → □□''P'' be an axiom in these systems? While the answer to this question is unclear, there is at least one axiom that is generally included in epistemic modal logic, because it is minimally true of all normal modal logics (see the section on axiomatic systems): * K, ''Distribution Axiom'': \Box (p \to q) \to (\Box p \to \Box q). It has been questioned whether the epistemic and alethic modalities should be considered distinct from each other. The criticism states that there is no real difference between "the truth in the world" (alethic) and "the truth in an individual's mind" (epistemic). An investigation has not found a single language in which alethic and epistemic modalities are formally distinguished, as by the means of a grammatical mood.


Temporal logic

Temporal logic is an approach to the semantics of expressions with tense, that is, expressions with qualifications of when. Some expressions, such as '2 + 2 = 4', are true at all times, while tensed expressions such as 'John is happy' are only true sometimes. In temporal logic, tense constructions are treated in terms of modalities, where a standard method for formalizing talk of time is to use ''two'' pairs of operators, one for the past and one for the future (P will just mean 'it is presently the case that P'). For example: :F''P'' : It will sometimes be the case that ''P'' :G''P'' : It will always be the case that ''P'' :P''P'' : It was sometime the case that ''P'' :H''P'' : It has always been the case that ''P'' There are then at least three modal logics that we can develop. For example, we can stipulate that, : \Diamond P = ''P'' is the case at some time ''t'' : \Box P = ''P'' is the case at every time ''t'' Or we can trade these operators to deal only with the future (or past). For example, : \Diamond_1 P = F''P'' : \Box_1 P = G''P'' or, : \Diamond_2 P = ''P'' and/or F''P'' : \Box_2 P = ''P'' and G''P'' The operators F and G may seem initially foreign, but they create normal modal systems. Note that F''P'' is the same as ¬G¬''P''. We can combine the above operators to form complex statements. For example, P''P'' → □P''P'' says (effectively), ''Everything that is past and true is necessary''. It seems reasonable to say that possibly it will rain tomorrow, and possibly it won't; on the other hand, since we can't change the past, if it is true that it rained yesterday, it cannot be true that it may not have rained yesterday. It seems the past is "fixed", or necessary, in a way the future is not. This is sometimes referred to as accidental necessity. But if the past is "fixed", and everything that is in the future will eventually be in the past, then it seems plausible to say that future events are necessary too. Similarly, the
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. The problem of future contingents seems to have been fi ...
considers the semantics of assertions about the future: is either of the propositions 'There will be a sea battle tomorrow', or 'There will not be a sea battle tomorrow' now true? Considering this thesis led
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...
to reject the
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. A logic satisfying this principle is called ...
for assertions concerning the future. Additional binary operators are also relevant to temporal logics (see Linear temporal logic). Versions of temporal logic can be used in
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
to model computer operations and prove theorems about them. In one version, ◇''P'' means "at a future time in the computation it is possible that the computer state will be such that P is true"; □''P'' means "at all future times in the computation P will be true". In another version, ◇''P'' means "at the immediate next state of the computation, ''P'' might be true"; □''P'' means "at the immediate next state of the computation, P will be true". These differ in the choice of
Accessibility relation An accessibility relation is a relation which plays a key role in assigning truth values to sentences in the relational semantics for modal logic. In relational semantics, a modal formula's truth value at a '' possible world'' w can depend on ...
. (P always means "P is true at the current computer state".) These two examples involve nondeterministic or not-fully-understood computations; there are many other modal logics specialized to different types of program analysis. Each one naturally leads to slightly different axioms.


Deontic logic

Likewise talk of morality, or of
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 ...
and norms generally, seems to have a modal structure. The difference between "You must do this" and "You may do this" looks a lot like the difference between "This is necessary" and "This is possible". Such logics are called '' deontic'', from the Greek for "duty". Deontic logics commonly lack the axiom T semantically corresponding to the reflexivity of the accessibility relation in
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 Jo ...
: in symbols, \Box\phi\to\phi. Interpreting □ as "it is obligatory that", T informally says that every obligation is true. For example, if it is obligatory not to kill others (i.e. killing is morally forbidden), then T implies that people actually do not kill others. The consequent is obviously false. Instead, using
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 Jo ...
, we say that though our own world does not realize all obligations, the worlds accessible to it do (i.e., T holds at these worlds). These worlds are called ''idealized'' worlds. ''P'' is obligatory with respect to our own world if at all idealized worlds accessible to our world, ''P'' holds. Though this was one of the first interpretations of the formal semantics, it has recently come under criticism. One other principle that is often (at least traditionally) accepted as a deontic principle is ''D'', \Box\phi\to\Diamond\phi, which corresponds to the seriality (or extendability or unboundedness) of the accessibility relation. It is an embodiment of the Kantian idea that "ought implies can". (Clearly the "can" can be interpreted in various senses, e.g. in a moral or alethic sense.)


Intuitive problems with deontic logic

When we try to formalize ethics with standard modal logic, we run into some problems. Suppose that we have a proposition ''K'': you have stolen some money, and another, ''Q'': you have stolen a small amount of money. Now suppose we want to express the thought that "if you have stolen some money, it ought to be a small amount of money". There are two likely candidates, : (1) (K \to \Box Q) : (2) \Box (K \to Q) But (1) and ''K'' together entail □''Q'', which says that it ought to be the case that you have stolen a small amount of money. This surely isn't right, because you ought not to have stolen anything at all. And (2) doesn't work either: If the right representation of "if you have stolen some money it ought to be a small amount" is (2), then the right representation of (3) "if you have stolen some money then it ought to be a large amount" is \Box (K \to (K \land \lnot Q)). Now suppose (as seems reasonable) that you ought not to steal anything, or \Box \lnot K. But then we can deduce \Box (K \to (K \land \lnot Q)) via \Box (\lnot K) \to \Box (K \to K \land \lnot K) and \Box (K \land \lnot K \to (K \land \lnot Q)) (the
contrapositive In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a statem ...
of Q \to K); so sentence (3) follows from our hypothesis (of course the same logic shows sentence (2)). But that can't be right, and is not right when we use natural language. Telling someone they should not steal certainly does not imply that they should steal large amounts of money if they do engage in theft.


Doxastic logic

''Doxastic logic'' concerns the logic of belief (of some set of agents). The term doxastic is derived from the
ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic p ...
''doxa'' which means "belief". Typically, a doxastic logic uses □, often written "B", to mean "It is believed that", or when relativized to a particular agent s, "It is believed by s that".


Metaphysical questions

In the most common interpretation of modal logic, one considers " logically possible worlds". If a statement is true in all
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 ...
, then it is a necessary truth. If a statement happens to be true in our world, but is not true in all possible worlds, then it is a contingent truth. A statement that is true in some possible world (not necessarily our own) is called a possible truth. Under this "possible worlds idiom," to maintain that Bigfoot's existence is possible but not actual, one says, "There is some possible world in which Bigfoot exists; but in the actual world, Bigfoot does not exist". However, it is unclear what this claim commits us to. Are we really alleging the existence of possible worlds, every bit as real as our actual world, just not actual?
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 em ...
believes that 'possible world' is something of a misnomer – that the term 'possible world' is just a useful way of visualizing the concept of possibility. For him, the sentences "you could have rolled a 4 instead of a 6" and "there is a possible world where you rolled a 4, but you rolled a 6 in the actual world" are not significantly different statements, and neither commit us to the existence of a possible world. David Lewis, on the other hand, made himself notorious by biting the bullet, asserting that all merely possible worlds are as real as our own, and that what distinguishes our world as ''actual'' is simply that it is indeed our world – ''
this This may refer to: * ''This'', the singular proximal demonstrative pronoun Places * This, or ''Thinis'', an ancient city in Upper Egypt * This, Ardennes, a commune in France People with the surname * Hervé This, French culinary chemist Arts, ...
'' world. That position is a major tenet of " modal realism". Some philosophers decline to endorse any version of modal realism, considering it ontologically extravagant, and prefer to seek various ways to paraphrase away these ontological commitments. Robert Adams holds that 'possible worlds' are better thought of as 'world-stories', or consistent sets of propositions. Thus, it is possible that you rolled a 4 if such a state of affairs can be described coherently. Computer scientists will generally pick a highly specific interpretation of the modal operators specialized to the particular sort of computation being analysed. In place of "all worlds", you may have "all possible next states of the computer", or "all possible future states of the computer".


Further applications

Modal logics have begun to be used in areas of the humanities such as literature, poetry, art and history.


History

The basic ideas of modal logic date back to antiquity.
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...
developed a modal syllogistic in Book I of his '' Prior Analytics'' (ch. 8–22), which
Theophrastus Theophrastus (; grc-gre, Θεόφραστος ; c. 371c. 287 BC), a Greek philosopher and the successor to Aristotle in the Peripatetic school. He was a native of Eresos in Lesbos.Gavin Hardy and Laurence Totelin, ''Ancient Botany'', Routle ...
attempted to improve. There are also passages in Aristotle's work, such as the famous sea-battle argument in '' De Interpretatione'' §9, that are now seen as anticipations of the connection of modal logic with potentiality and time. In the Hellenistic period, the logicians
Diodorus Cronus Diodorus Cronus ( el, Διόδωρος Κρόνος; died c. 284 BC) was a Greek philosopher and dialectician connected to the Megarian school. He was most notable for logic innovations, including his master argument formulated in response to A ...
,
Philo the Dialectician Philo the Dialectician ( el, Φίλων; fl. 300 BC) was a Greek philosopher of the Megarian (Dialectical) school. He is sometimes called Philo of Megara although the city of his birth is unknown. He is most famous for the debate he had with his ...
and the Stoic Chrysippus each developed a modal system that accounted for the interdefinability of possibility and necessity, accepted axiom T (see below), and combined elements of modal logic and
temporal logic In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time (for example, "I am ''always'' hungry", "I will ''eventually'' be hungry", or "I will be hungry ''until'' I ...
in attempts to solve the notorious
Master Argument :''See Diodorus Cronus § Master argument for the classical master argument related to the problem of future contingents.'' The master argument is George Berkeley's argument that mind-independent objects do not exist because it is impossible to ...
. The earliest formal system of modal logic was developed by Avicenna, who ultimately developed a theory of " temporally modal" syllogistic.History of logic: Arabic logic
''
Encyclopædia Britannica The (Latin for "British Encyclopædia") is a general knowledge English-language encyclopaedia. It is published by Encyclopædia Britannica, Inc.; the company has existed since the 18th century, although it has changed ownership various t ...
''.
Modal logic as a self-aware subject owes much to the writings of the
Scholastics Scholasticism was a medieval school of philosophy that employed a critical organic method of philosophical analysis predicated upon the Aristotelian 10 Categories. Christian scholasticism emerged within the monastic schools that translate ...
, in particular
William of Ockham William of Ockham, OFM (; also Occam, from la, Gulielmus Occamus; 1287 – 10 April 1347) was an English Franciscan friar, scholastic philosopher, apologist, and Catholic theologian, who is believed to have been born in Ockham, a small vil ...
and
John Duns Scotus John Duns Scotus ( – 8 November 1308), commonly called Duns Scotus ( ; ; "Duns the Scot"), was a Scottish Catholic priest and Franciscan friar, university professor, philosopher, and theologian. He is one of the four most important ...
, who reasoned informally in a modal manner, mainly to analyze statements about
essence Essence ( la, essentia) is a polysemic term, used in philosophy and theology as a designation for 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 ...
and
accident An accident is an unintended, normally unwanted event that was not directly caused by humans. The term ''accident'' implies that nobody should be blamed, but the event may have been caused by unrecognized or unaddressed risks. Most researche ...
. In the 19th century, Hugh MacColl made innovative contributions to modal logic, but did not find much acknowledgment.
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 ...
founded modern modal logic in a series of scholarly articles beginning in 1912 with "Implication and the Algebra of Logic". Lewis was led to invent modal logic, and specifically strict implication, on the grounds that classical logic grants paradoxes of material implication such as the principle that a falsehood implies any proposition. This work culminated in his 1932 book ''Symbolic Logic'' (with C. H. Langford), which introduced the five systems ''S1'' through ''S5''. After Lewis, modal logic received little attention for several decades.
Nicholas Rescher Nicholas Rescher (; ; born 15 July 1928) is a German-American philosopher, polymath, and author, who has been a professor of philosophy at the University of Pittsburgh since 1961. He is chairman of the Center for Philosophy of Science and was fo ...
has argued that this was because
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
rejected it. However, Jan Dejnozka has argued against this view, stating that a modal system which Dejnozka calls "MDL" is described in Russell's works, although Russell did believe the concept of modality to "come from confusing propositions with
propositional function In propositional calculus, a propositional function or a predicate is a sentence expressed in a way that would assume the value of true or false, except that within the sentence there is a variable (''x'') that is not defined or specified (thus be ...
s," as he wrote in ''The Analysis of Matter''. Arthur Norman Prior warned
Ruth Barcan Marcus Ruth Barcan Marcus (; born Ruth Charlotte Barcan; 2 August 1921 – 19 February 2012) was an American academic philosopher and logician best known for her work in modal and philosophical logic. She developed the first formal systems of quant ...
to prepare well in the debates concerning quantified modal logic with
Willard Van Orman Quine Willard Van Orman Quine (; known to his friends 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" ...
, due to the biases against modal logic. Ruth C. Barcan (later
Ruth Barcan Marcus Ruth Barcan Marcus (; born Ruth Charlotte Barcan; 2 August 1921 – 19 February 2012) was an American academic philosopher and logician best known for her work in modal and philosophical logic. She developed the first formal systems of quant ...
) developed the first axiomatic systems of quantified modal logic — first and second order extensions of Lewis' ''S2'', ''S4'', and ''S5''. The contemporary era in modal semantics began in 1959, when
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 em ...
(then only a 18-year-old
Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of high ...
undergraduate) introduced the now-standard
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 Jo ...
for modal logics. These are commonly referred to as "possible worlds" semantics. Kripke and A. N. Prior had previously corresponded at some length. Kripke semantics is basically simple, but proofs are eased using semantic-tableaux or
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. An analytic tableau is a tree structure compu ...
, as explained by E. W. Beth. A. N. Prior created modern
temporal logic In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time (for example, "I am ''always'' hungry", "I will ''eventually'' be hungry", or "I will be hungry ''until'' I ...
, closely related to modal logic, in 1957 by adding modal operators and meaning "eventually" and "previously". Vaughan Pratt introduced dynamic logic in 1976. In 1977, Amir Pnueli proposed using temporal logic to formalise the behaviour of continually operating concurrent programs. Flavors of temporal logic include propositional dynamic logic (PDL), (propositional) linear temporal logic (LTL),
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 realiz ...
(CTL),
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. It was introduced in 1980 by Matthew Hennessy and Robin Milner in their pap ...
, and ''T''. The mathematical structure of modal logic, namely
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
s augmented with
unary operation In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...
s (often called modal algebras), began to emerge with J. C. C. McKinsey's 1941 proof that ''S2'' and ''S4'' are decidable, and reached full flower in the work of
Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
and his student Bjarni Jónsson (Jónsson and Tarski 1951–52). This work revealed that ''S4'' and ''S5'' are models of interior algebra, a proper extension of Boolean algebra originally designed to capture the properties of the interior and
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 that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are de ...
s of
topology 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 ...
. Texts on modal logic typically do little more than mention its connections with the study of
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
s and
topology 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 ...
. For a thorough survey of the history of formal modal logic and of the associated mathematics, see
Robert Goldblatt __notoc__ Robert Ian Goldblatt (born 1949) is a mathematical logician who is Emeritus Professor in the School of Mathematics and Statistics at Victoria University, Wellington, New Zealand. His most popular books are ''Logics of Time and Computatio ...
(2006).Robert Goldbaltt
Mathematical Modal Logic: A view of it evolution
/ref>


See also

*
Accessibility relation An accessibility relation is a relation which plays a key role in assigning truth values to sentences in the relational semantics for modal logic. In relational semantics, a modal formula's truth value at a '' possible world'' w can depend on ...
* Conceptual necessity *
Counterpart theory In philosophy, specifically in the area of metaphysics, counterpart theory is an alternative to standard ( Kripkean) possible-worlds semantics for interpreting quantified modal logic. Counterpart theory still presupposes possible worlds, but diffe ...
*
David Kellogg Lewis David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". w ...
* ''De dicto'' and ''de re'' * Description logic *
Doxastic logic Doxastic logic is a type of logic concerned with reasoning about beliefs. The term ' derives from the Ancient Greek (''doxa'', "opinion, belief"), from which the English term '' doxa'' ("popular opinion or belief") is also borrowed. Typically, a ...
* Dynamic logic * Enthymeme *
Free choice inference Free choice is a phenomenon in natural language where a linguistic disjunction appears to receive a logical conjunctive interpretation when it interacts with a modal operator. For example, the following English sentences can be interpreted to mean ...
*
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. In formal logic, there; is a trade-off between expressiveness and computational tractability. The hist ...
* Interior algebra * Interpretability logic *
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 Jo ...
*
Metaphysical necessity In philosophy, metaphysical necessity, sometimes called broad logical necessity, is one of many different kinds of necessity, which sits between logical necessity and nomological (or physical) necessity, in the sense that logical necessity entails ...
*
Modal verb A modal verb is a type of verb that contextually indicates a modality such as a ''likelihood'', ''ability'', ''permission'', ''request'', ''capacity'', ''suggestion'', ''order'', ''obligation'', or ''advice''. Modal verbs generally accompany the b ...
* Multimodal logic *
Multi-valued logic Many-valued logic (also multi- or multiple-valued logic) refers to a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false ...
*
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 ...
*
Provability logic Provability logic is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic. Examples ...
*
Regular modal logic In modal logic, a regular modal logic is a modal logic containing (as axiom or theorem) the duality of the modal operators: \Diamond A \leftrightarrow \lnot\Box\lnot A and closed under the rule \frac. Every normal modal logic In logic, a norma ...
*
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. They may be viewed as a family of substructural or modal logics. It is generally, but ...
*
Strict conditional In logic, a strict conditional (symbol: \Box, or ⥽) is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necess ...
* Two-dimensionalism


Notes


References

* ''This article includes material from the''
Free On-line Dictionary of Computing The Free On-line Dictionary of Computing (FOLDOC) is an online, searchable, encyclopedic dictionary of computing subjects. History FOLDOC was founded in 1985 by Denis Howe and was hosted by Imperial College London. In May 2015, the site was up ...
, ''used with permission under the''
GFDL 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. It is similar to the GNU General Public License, giving readers the ...
. * Barcan-Marcus, Ruth JSL 11 (1946) and JSL 112 (1947) and "Modalities", OUP, 1993, 1995. * Beth, Evert W., 1955.
Semantic entailment and formal derivability
, Mededlingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde, N.R. Vol 18, no 13, 1955, pp 309–42. Reprinted in Jaakko Intikka (ed.) The Philosophy of Mathematics, Oxford University Press, 1969 (Semantic Tableaux proof methods). * Beth, Evert W.,
Formal Methods: An Introduction to Symbolic Logic and to the Study of Effective Operations in Arithmetic and Logic
, D. Reidel, 1962 (Semantic Tableaux proof methods). * Blackburn, P.; van Benthem, J.; and Wolter, Frank; Eds. (2006)
Handbook of Modal Logic
'. North Holland. * Blackburn, Patrick; de Rijke, Maarten; and Venema, Yde (2001) ''Modal Logic''. Cambridge University Press. * Chagrov, Aleksandr; and Zakharyaschev, Michael (1997) ''Modal Logic''. Oxford University Press. * Chellas, B. F. (1980)
Modal Logic: An Introduction
'. Cambridge University Press. * Cresswell, M. J. (2001) "Modal Logic" in Goble, Lou; Ed., ''The Blackwell Guide to Philosophical Logic''. Basil Blackwell: 136–58. * Fitting, Melvin; and Mendelsohn, R. L. (1998) ''First Order Modal Logic''. Kluwer. * James Garson (2006) ''Modal Logic for Philosophers''. Cambridge University Press. . A thorough introduction to modal logic, with coverage of various derivation systems and a distinctive approach to the use of diagrams in aiding comprehension. * Girle, Rod (2000) ''Modal Logics and Philosophy''. Acumen (UK). . Proof by refutation trees. A good introduction to the varied interpretations of modal logic.
Goldblatt, Robert
(1992) "Logics of Time and Computation", 2nd ed., CSLI Lecture Notes No. 7. University of Chicago Press. * —— (1993) ''Mathematics of Modality'', CSLI Lecture Notes No. 43. University of Chicago Press. * —— (2006)
Mathematical Modal Logic: a View of its Evolution
, in Gabbay, D. M.; and Woods, John; Eds., ''Handbook of the History of Logic, Vol. 6''. Elsevier BV. * Goré, Rajeev (1999) "Tableau Methods for Modal and Temporal Logics" in D'Agostino, M.; Gabbay, D.; Haehnle, R.; and Posegga, J.; Eds., ''Handbook of Tableau Methods''. Kluwer: 297–396. * Hughes, G. E., and Cresswell, M. J. (1996) ''A New Introduction to Modal Logic''. Routledge. * Jónsson, B. and Tarski, A., 1951–52, "Boolean Algebra with Operators I and II", ''American Journal of Mathematics 73'': 891–939 and ''74'': 129–62. * Kracht, Marcus (1999)
Tools and Techniques in Modal Logic
', Studies in Logic and the Foundations of Mathematics No. 142. North Holland. * Lemmon, E. J. (with Scott, D.) (1977) ''An Introduction to Modal Logic'', American Philosophical Quarterly Monograph Series, no. 11 (Krister Segerberg, series ed.). Basil Blackwell. * Lewis, C. I. (with Langford, C. H.) (1932). ''Symbolic Logic''. Dover reprint, 1959. * Prior, A. N. (1957)
Time and Modality
'. Oxford University Press. * Snyder, D. Paul "Modal Logic and its applications", Van Nostrand Reinhold Company, 1971 (proof tree methods). * Zeman, J. J. (1973)
Modal Logic.
' Reidel. Employs
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 ...
.
"History of logic"
Britannica Online.


Further reading

* Ruth Barcan Marcus, ''Modalities'', Oxford University Press, 1993. * D. M. Gabbay, A. Kurucz, F. Wolter and M. Zakharyaschev, ''Many-Dimensional Modal Logics: Theory and Applications'', Elsevier, Studies in Logic and the Foundations of Mathematics, volume 148, 2003, . overs many varieties of modal logics, e.g. temporal, epistemic, dynamic, description, spatial from a unified perspective with emphasis on computer science aspects, e.g. decidability and complexity.* Andrea Borghini
''A Critical Introduction to the Metaphysics of Modality''
New York: Bloomsbury, 2016.


External links

* Internet Encyclopedia of Philosophy: **
Modal Logic: A Contemporary View
– by Johan van Benthem. **
Rudolf Carnap's Modal Logic
– by MJ Cresswell. * Stanford Encyclopedia of Philosophy: **
Modal Logic
– by James Garson. **
Modern Origins of Modal Logic
– by Roberta Ballarin. **
Provability Logic
– by Rineke Verbrugge. * Edward N. Zalta, 1995,
Basic Concepts in Modal Logic.
* John McCarthy, 1996,
Modal Logic.

Molle
a Java prover for experimenting with modal logics * Suber, Peter, 2002,



List of many modal logics with sources, by John Halleck.
Advances in Modal Logic.
Biannual international conference and book series in modal logic.
S4prover
A tableaux prover for S4 logic *
Some Remarks on Logic and Topology
– by Richard Moot; exposits a
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 ...
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 comp ...
for the modal logic S4.
LoTREC
The most generic prover for modal logics from IRIT/Toulouse University {{Authority control Logic Philosophical logic Mathematical logic Semantics