Modal logic is a collection of

^{0}''.
Other well-known elementary axioms are:
*4: $\backslash Box\; p\; \backslash to\; \backslash Box\; \backslash Box\; p$
*B: $p\; \backslash to\; \backslash Box\; \backslash Diamond\; p$
*D: $\backslash Box\; p\; \backslash to\; \backslash Diamond\; p$
*5: $\backslash Diamond\; p\; \backslash to\; \backslash Box\; \backslash Diamond\; p$
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

Saul Kripke
Saul Aaron Kripke (; born November 13, 1940) is an American philosopher
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the United States
** Americans, citizens and nationa ...

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.

''Encyclopædia Britannica''. Modal logic as a self-aware subject owes much to the writings of the Scholastics, in particular William of Ockham and John Duns Scotus, who reasoned informally in a modal manner, mainly to analyze statements about essence and accident (philosophy), accident. In the 19th century, Hugh MacColl made innovative contributions to modal logic, but did not find much acknowledgment. C. I. Lewis 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 Vacuous truth, a falsehood implies any proposition. This work culminated in his 1932 book ''Symbolic Logic'' (with Cooper Harold Langford, C. H. Langford), which introduced the five systems ''S1'' through ''S5''. After Lewis, modal logic received little attention for several decades. Nicholas Rescher has argued that this was because Bertrand Russell 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 functions," as he wrote in ''The Analysis of Matter''. Arthur Norman Prior warned Ruth Barcan Marcus to prepare well in the debates concerning quantified modal logic with Willard Van Orman Quine, due to the biases against modal logic. Ruth C. Barcan (later Ruth Barcan Marcus) 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, whenSaul Kripke
Saul Aaron Kripke (; born November 13, 1940) is an American philosopher
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the United States
** Americans, citizens and nationa ...

(then only a 18-year-old Harvard University 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 Joya ...

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, as explained by Evert Willem Beth, E. W. Beth.
A. N. Prior created modern temporal logic, closely related to modal logic, in 1957 by adding modal operators [F] and [P] meaning "eventually" and "previously". Vaughan Pratt introduced dynamic logic (modal logic), 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 (PLTL), linear temporal logic (LTL), computation tree logic (CTL), Hennessy–Milner logic, and ''T''.
The mathematical structure of modal logic, namely Boolean algebra (structure), Boolean algebras augmented with unary operations (often called modal algebras), began to emerge with J.C.C. McKinsey, J. C. C. McKinsey's 1941 proof that ''S2'' and ''S4'' are decidable, and reached full flower in the work of Alfred Tarski 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 operator, interior and closure operators of topology. Texts on modal logic typically do little more than mention its connections with the study of Boolean algebra (structure), Boolean algebras and topology. For a thorough survey of the history of formal modal logic and of the associated mathematics, see Robert Goldblatt (2006).Robert Goldbaltt

Mathematical Modal Logic: A view of it evolution

/ref>

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 Joya ...

* Metaphysical necessity
* Modal verb
* Multimodal logic
* Multi-valued logic
* Neighborhood semantics
* Provability logic
* Regular modal logic
* Relevance logic
* Strict conditional
* Two-dimensionalism

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.; Johan van Benthem (logician), 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. * Max Cresswell, 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 analytic tableau, 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. * Bjarni Jónsson, Jónsson, B. and Alfred Tarski, 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. * John Lemmon, Lemmon, E. J. (with Dana Scott, Scott, D.) (1977) ''An Introduction to Modal Logic'', American Philosophical Quarterly Monograph Series, no. 11 (Krister Segerberg, series ed.). Basil Blackwell. * Clarence Irving Lewis, Lewis, C. I. (with Cooper Harold Langford, Langford, C. H.) (1932). ''Symbolic Logic''. Dover reprint, 1959. * Arthur Prior, 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.

"History of logic"

Britannica Online.

''A Critical Introduction to the Metaphysics of Modality''

New York: Bloomsbury, 2016.

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 (computer scientist), 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 topology, topological semantics for the modal logic S4.

LoTREC

The most generic prover for modal logics from IRIT/Toulouse University {{Authority control Logic Modal logic, Philosophical logic Mathematical logic Semantics

formal system
A formal system is an 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 formal system is essen ...

s originally developed and still widely used to represent statements about necessity and possibility. The basic unary (1-place) modal operators are most often interpreted "□" for "Necessarily" and "◇" for "Possibly".
In a classical modal logic
Classical may refer to:
European antiquity
*Classical antiquity, a period of history from roughly the 7th or 8th century B.C.E. to the 5th century C.E. centered on the Mediterranean Sea
*Classical architecture, architecture derived from Greek and ...

, each can be expressed in terms of the other and negation
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, ...

in a De Morgan dual#Redirect DE {{Redirect category shell, 1=
{{Redirect from other capitalisation
{{Redirect from ambiguous term
...

ity:
:$\backslash Diamond\; P\; \backslash leftrightarrow\; \backslash lnot\; \backslash Box\; \backslash lnot\; P,\; \backslash quad\backslash quad\; \backslash Box\; P\; \backslash leftrightarrow\; \backslash lnot\; \backslash Diamond\; \backslash lnot\; P$
The modal formula $\backslash Box\; P\; \backslash rightarrow\; \backslash Diamond\; P$ can be read using the above interpretation as "if P is necessary, then it is also possible", which is almost always held to be valid. This interpretation of the modal operators as necessity and possibility is called alethic modal logic
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 ...

. There are modal logics of other modes, such as "□" for "Obligatorily" and "◇" for "Permissibly" in deontic modal logic, where this same formulae means "if P is obligatory, then it is permissible", which is also almost always held to be valid.
The first modal axiomatic system
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s were developed by C. I. Lewis in 1912, building on an informal tradition stretching back to Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher
A philosopher is someone who practices philosophy
Philosophy (from , ) is the study of general and fundamental questio ...

. The relational 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 Joy ...

for modal logic was developed 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
A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'l ...

, and Saul Kripke
Saul Aaron Kripke (; born November 13, 1940) is an American philosopher
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the United States
** Americans, citizens and nationa ...

in the mid twentieth century. In this semantics, 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. They are widely used as a formal device in logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize ...

''. A formula's truth value at one possible world can depend on the truth values of other formulas at other ''accessible'' possible worlds
A possible world is a complete and consistent way the world is or could have been. They are widely used as a formal device in logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize ...

. In particular, possibility amounts to truth at ''some'' accessible possible world while necessity amounts to truth at ''every'' accessible possible world.
Modal logic is often referred to as "the logic of necessity and possibility", and such applications continue to play a major role in philosophy of language
In analytic philosophy
Analytic philosophy is a branch and tradition of philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, ...

, epistemology
Epistemology (; ) is the Outline of philosophy, branch of philosophy concerned with knowledge. Epistemologists study the nature, origin, and scope of knowledge, epistemic Justification (epistemology), justification, the Reason, rationality o ...

, metaphysics
Metaphysics is the branch of philosophy that studies the first principles of being, identity and change, space and time, causality, necessity and possibility. It includes questions about the nature of consciousness and the relationship between ...

, and formal semantics. However, the mathematical apparatus of modal logic has proved useful in numerous other fields including game theory
Game theory is the study of mathematical model
A mathematical model is a description of a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
...

, moral
A moral (from Latin ''morālis'') is a message that is conveyed or a lesson to be learned from a narrative, story or wikt:event, event. The moral may be left to the hearer, reader, or viewer to determine for themselves, or may be explicitly enca ...

and legal theory
Jurisprudence, or legal theory, is the theoretical study of the propriety of law
Law is a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
...

, web design
Web design encompasses many different skills and disciplines in the production and maintenance of website
A website (also written as web site) is a collection of web page
A web page (or webpage) is a hypertext
File:Douglas Engelbar ...

, multiverse-based set theory, and social epistemology
Social epistemology refers to a broad set of approaches that can be taken in epistemology
Epistemology (; ) is the Outline of philosophy, branch of philosophy concerned with knowledge. Epistemologists study the nature, origin, and scope of kno ...

. One prominent textbook on the model theory of modal logic suggests that it can be seen more generally as the study of formal systems which take a local perspective on relational structures.
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. They are widely used as a formal device in logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize ...

''. For a formula that contains a modal operator, its truth value can depend on what is true at other accessible
Accessibility in the sense considered here refers to the design of products, devices, services, or environments so as to be usable by people with disabilities. The concept of accessible design and practice of accessible development ensures b ...

worlds. Thus, the relational semantics interprets formulas of modal logic using models defined as follows.
* A ''relational model'' is a tuple $\backslash mathfrak\; =\; \backslash langle\; W,\; R,\; V\; \backslash rangle$ 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\; \backslash times\; F\; \backslash to\; \backslash $ 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 Kripke semantics, relational semantics for modal logic. In relational semantics, a modal formula's truth value at a ''possible world'' w ...

, 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. It determines which atomic formula
In mathematical logic
Mathematical logic, also called formal logic, is a subfield of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alg ...

s are true at which worlds.
Then we recursively define the truth of a formula at a world in a model $\backslash mathfrak$:
* $\backslash mathfrak,\; w\; \backslash models\; P$ iff $V(w,\; P)=1$
* $\backslash mathfrak,\; w\; \backslash models\; \backslash neg\; P$ iff $w\; \backslash not\; \backslash models\; P$
* $\backslash mathfrak,\; w\; \backslash models\; (P\; \backslash wedge\; Q)$ iff $w\; \backslash models\; P$ and $w\; \backslash models\; Q$
* $\backslash mathfrak,\; w\; \backslash models\; \backslash Box\; P$ iff for every element $u$ of $W$, if $w\; R\; u$ then $u\; \backslash models\; P$
* $\backslash mathfrak,\; w\; \backslash models\; \backslash Diamond\; P$ iff for some element $u$ of $W$, it holds that $w\; R\; u$ and $u\; \backslash models\; P$
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 $\backslash mathfrak$ whose accessibility relation is reflexive. Because the relation is reflexive, we will have that $\backslash mathfrak,w\; \backslash models\; P\; \backslash rightarrow\; \backslash Diamond\; P$ for any $w\; \backslash in\; G$ 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 $\backslash mathfrak\; =\; \backslash langle\; G,\; R\; \backslash rangle$ 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 Greek συμμετρία ''symmetria'' "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 pre ...

if ''w R u'' implies ''u R w'', for all ''w'' and ''u'' in ''G''
* transitive
Transitivity or transitive may refer to:
Grammar
* Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects
* Transitive verb, a verb which takes an object
* Transitive case, a grammatical case to mark arg ...

if ''w R u'' and ''u R q'' together imply ''w R q'', for all ''w'', ''u'', ''q'' in ''G''.
* serial
Serial may refer to:
Arts, entertainment, and media The presentation of works in sequential segments
* Serial (literature), serialised fiction in print
* Serial (publishing), periodical publications and newspapers
* Serial (radio and television), ...

if, for every ''w'' in ''G'' there is some ''u'' in ''G'' such that ''w R u''.
* Euclidean
Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. It is the name of:
Geometry
*Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry a ...

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'')
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
Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. It is the name of:
Geometry
*Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry a ...

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 Greek συμμετρία ''symmetria'' "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 pre ...

and transitive
Transitivity or transitive may refer to:
Grammar
* Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects
* Transitive verb, a verb which takes an object
* Transitive case, a grammatical case to mark arg ...

as well. Hence for models of S5, ''R'' is an equivalence relation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, 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:
: $w\; \backslash models\; \backslash Diamond\; P$ if and only if for some element ''u'' of ''G'', it holds that $u\; \backslash models\; P$ and ''w R u''.
If we consider frames based on the total relation we can just say that
: $w\; \backslash models\; \backslash Diamond\; P$ if and only if for some element ''u'' of ''G'', it holds that $u\; \backslash models\; P$.
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 $P\; \backslash implies\; \backslash Box\backslash Diamond\; P$, $\backslash Box\; P\; \backslash implies\; \backslash Box\backslash Box\; P$ and $\backslash Box\; P\; \backslash implies\; P$ (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 $\backslash Chi\; =\; \backslash langle\; X,\; \backslash tau,\; V\; \backslash rangle$ where $\backslash langle\; X,\; \backslash tau\; \backslash rangle$ is atopological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

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:
* $\backslash Chi,\; x\; \backslash models\; P$ iff $x\; \backslash in\; V(p)$
* $\backslash Chi,\; x\; \backslash models\; \backslash neg\; \backslash phi$ iff $\backslash Chi,\; x\; \backslash not\backslash models\; \backslash phi$
* $\backslash Chi,\; x\; \backslash models\; \backslash phi\; \backslash land\; \backslash chi$ iff $\backslash Chi,\; x\; \backslash models\; \backslash phi$ and $\backslash Chi,\; x\; \backslash models\; \backslash chi$
* $\backslash Chi,\; x\; \backslash models\; \backslash Box\; \backslash phi$ iff for some $U\; \backslash in\; \backslash tau$ we have both that $x\; \backslash in\; U$ and also that $\backslash Chi,\; y\; \backslash models\; \backslash phi$ for all $y\; \backslash in\; U$
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 LewisDavid or Dave Lewis may refer to:
Academics
*A. David Lewis (born 1977), American comic writer and scholar of religion and literature
*David Lewis (academic) (born 1960), English scholar of development
*David Lewis (lawyer) ( – 1584), Welsh ...

and Angelika Kratzer
Angelika Kratzer is a professor emerita of linguistics in the department of linguistics at the University of Massachusetts Amherst.
Biography
She was born in Germany, and received her PhD from the University of Konstanz in 1979, with a dissertati ...

's logics for counterfactuals
Counterfactual conditionals (also ''subjunctive'' or ''X-marked'') are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactual ...

.
Axiomatic systems

The first formalizations of modal logic wereaxiomatic
An axiom, postulate or assumption is a statement that is taken to be true
True most commonly refers to truth
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language ...

. Numerous variations with very different properties have been proposed since C. I. Lewis began working in the area in 1912. Hughes and Cresswell (1996), for example, describe 42 normal 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
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fal ...

with two unary operations, one denoting "necessity" and the other "possibility". The notation of C. I. Lewis, 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''". 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 the field of functional analysis
Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction i ...

of operators.
In many modal logics, the necessity and possibility operators satisfy the following analogues of de Morgan's laws
In propositional calculus, propositional logic and Boolean algebra, De Morgan's laws are a pair of transformation rules that are both Validity (logic), valid rule of inference, rules of inference. They are named after Augustus De Morgan, a 19th ...

from Boolean algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

:
:"It is not necessary that ''X''" is logically equivalent
Logic (from Greek: grc, λογική, label=none, lit=possessed of reason
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=n ...

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
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fal ...

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 logicIn logic, a normal modal logic is a set ''L'' of modal formulas such that ''L'' contains:
* All propositional tautology (logic), tautologies;
* All instances of the Saul_Kripke, Kripke schema: \Box(A\to B)\to(\Box A\to\Box B)
and it is closed under:
...

s, include the following rule and axiom:
* N, Necessitation Rule: If ''p'' is a theorem
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

(of any system invoking N), then □''p'' is likewise a theorem.
* K, Distribution Axiom:
The weakest normal modal logicIn logic, a normal modal logic is a set ''L'' of modal formulas such that ''L'' contains:
* All propositional tautology (logic), tautologies;
* All instances of the Saul_Kripke, Kripke schema: \Box(A\to B)\to(\Box A\to\Box B)
and it is closed under:
...

, named "K" in honor of Saul Kripke
Saul Aaron Kripke (; born November 13, 1940) is an American philosopher
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the United States
** Americans, citizens and nationa ...

, is simply the propositional calculus
Propositional calculus is a branch of logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fal ...

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 ''S1normal modal logicIn logic, a normal modal logic is a set ''L'' of modal formulas such that ''L'' contains:
* All propositional tautology (logic), tautologies;
* All instances of the Saul_Kripke, Kripke schema: \Box(A\to B)\to(\Box A\to\Box B)
and it is closed under:
...

. But specific rules or sets of rules may be appropriate for specific systems. For example, in deontic logicDeontic logic is the field of philosophical logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logo ...

, $\backslash Box\; p\; \backslash to\; \backslash Diamond\; p$ (If it ought to be that ''p'', then it is permitted that ''p'') seems appropriate, but we should probably not include that $p\; \backslash to\; \backslash Box\; \backslash Diamond\; p$. In fact, to do so is to commit the appeal to nature
An appeal to nature is an argument
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc ...

fallacy (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 ofanalytic proofIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

). More complex calculi have been applied to modal logic to achieve generality.
Decision methods

Analytic tableaux
In proof theory
Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Jon Barwise, Barwise (1978) consists o ...

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 theLatin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became ...

''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, epistemic
Epistemology (; ) is the branch of philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, ...

, and so on, than it is to make sense of relativizing other notions.
In classical modal logic
Classical may refer to:
European antiquity
*Classical antiquity, a period of history from roughly the 7th or 8th century B.C.E. to the 5th century C.E. centered on the Mediterranean Sea
*Classical architecture, architecture derived from Greek and ...

, 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 logic
Propositional calculus is a branch of logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ) ...

. 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
Athens ( ; el, Αθήνα, Athína ; grc, Ἀθῆναι, Athênai (pl.) ) is the capital city, capital and List of cities in Greece, largest city of Greece. Athens domi ...

(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 with 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 ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...

) 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 Joya ...

.
Physical possibility

Something is physically, or nomically, possible if it is permitted by thelaws of physics
Scientific laws or laws of science are statements, based on repeated experiment
An experiment is a procedure carried out to support, refute, or validate a hypothesis. Experiments provide insight into Causality, cause-and-effect by demonstrat ...

. For example, current theory is thought to allow for there to be an atom
An atom is the smallest unit of ordinary matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of ato ...

with an atomic number
The atomic number or proton number (symbol ''Z'') of a chemical element
In chemistry, an element is a pure Chemical substance, substance consisting only of atoms that all have the same numbers of protons in their atomic nucleus, nuclei. ...

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
A vacuum is a space
Space is the boundless three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...

, 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 someone who practices philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, ...

debate if objects have properties independent of those dictated by scientific laws. For example, it might be metaphysically necessary, as some who advocate physicalism
In philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about existence
Existence is the ability of an entity to interact with physical reality
Reality is the sum or aggregate of all that is ...

have thought, that all thinking beings have bodies and can experience the passage of time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

. Saul Kripke
Saul Aaron Kripke (; born November 13, 1940) is an American philosopher
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the United States
** Americans, citizens and nationa ...

has argued that every person necessarily has the parents they do have: anyone with different parents would not be the same person.
Metaphysical possibility 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 thatBigfoot
Bigfoot, also commonly referred to as Sasquatch, in Canadian folklore, Canadian and Folklore of the United States, American folklore, is an ape-like creature that is purported to inhabit the forests of North America. Supposed evidence of 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
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician ...

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
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician ...

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'': $\backslash Box\; (p\; \backslash to\; q)\; \backslash to\; (\backslash Box\; p\; \backslash to\; \backslash Box\; q)$.
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
In linguistics
Linguistics is the scientific study of language
A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) and writing. Most langua ...

.
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, :$\backslash Diamond\; P$ = ''P'' is the case at some time ''t'' :$\backslash Box\; P$ = ''P'' is the case at every time ''t'' Or we can trade these operators to deal only with the future (or past). For example, :$\backslash Diamond\_1\; P$ = F''P'' :$\backslash Box\_1\; P$ = G''P'' or, :$\backslash Diamond\_2\; P$ = ''P'' and/or F''P'' :$\backslash Box\_2\; P$ = ''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 probably isn't 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, theproblem of future contingents
Future contingent propositions (or simply, future contingents) are statements about states of affairs in the future that are ''contingent
In philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those ...

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
A philosopher is someone who practices philosophy
Philosophy (from , ) is the study of general and fundamental questio ...

to reject the principle of bivalence
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label ...

for assertions concerning the future.
Additional binary operators are also relevant to temporal logics, ''q.v.'' Linear Temporal LogicIn logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, acc ...

.
Versions of temporal logic can be used in computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of computation, automation, a ...

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 Kripke semantics, relational semantics for modal logic. In relational semantics, a modal formula's truth value at a ''possible world'' w ...

. (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 ofobligation
An obligation is a course of action that someone is required to take, whether legal
Law is a system
A system is a group of interacting
Interaction is a kind of action that occurs as two or more objects have an effect upon one anothe ...

and norms
Norm, the Norm or NORM may refer to:
In academic disciplines
* Norm (geology), an estimate of the idealised mineral content of a rock
* Norm (philosophy), a standard in normative ethics that is prescriptive rather than a descriptive or explanato ...

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 Joya ...

: in symbols, $\backslash Box\backslash phi\backslash to\backslash phi$. 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 Joya ...

, 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'', $\backslash Box\backslash phi\backslash to\backslash Diamond\backslash phi$, 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) $(K\; \backslash to\; \backslash Box\; Q)$ : (2) $\backslash Box\; (K\; \backslash to\; Q)$ 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 $\backslash Box\; (K\; \backslash to\; (K\; \backslash land\; \backslash lnot\; Q))$. Now suppose (as seems reasonable) that you ought not to steal anything, or $\backslash Box\; \backslash lnot\; K$. But then we can deduce $\backslash Box\; (K\; \backslash to\; (K\; \backslash land\; \backslash lnot\; Q))$ via $\backslash Box\; (\backslash lnot\; K)\; \backslash to\; \backslash Box\; (K\; \backslash to\; K\; \backslash land\; \backslash lnot\; K)$ and $\backslash Box\; (K\; \backslash land\; \backslash lnot\; K\; \backslash to\; (K\; \backslash land\; \backslash lnot\; Q))$ (thecontrapositive
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...

of $Q\; \backslash to\; K$); 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 theancient Greek
Ancient Greek includes the forms of the Greek language
Greek ( el, label=Modern Greek
Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the diale ...

''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 allpossible worlds
A possible world is a complete and consistent way the world is or could have been. They are widely used as a formal device in logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize ...

, 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? David LewisDavid or Dave Lewis may refer to:
Academics
*A. David Lewis (born 1977), American comic writer and scholar of religion and literature
*David Lewis (academic) (born 1960), English scholar of development
*David Lewis (lawyer) ( – 1584), Welsh ...

, 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 (Egypt), This, or ''Thinis'', an ancient city in Upper Egypt
* This, Ardennes, a commune in France
People with the surname
* Hervé This, French culinary c ...

'' world. That position is a major tenet of "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; pos ...

". 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
A philosopher is someone who practices philosophy
Philosophy (from , ) is the study of general and fundamental questio ...

developed a modal syllogistic in Book I of his ''Prior Analytics
The ''Prior Analytics'' ( grc-gre, Ἀναλυτικὰ Πρότερα; la, Analytica Priora) is a work by Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher
A p ...

'' (chs 8–22), which Theophrastus
Theophrastus (; grc-gre, Θεόφραστος ; c. 371c. 287 BC), a Greek native of Eresos in Lesbos Island, Lesbos,Gavin Hardy and Laurence Totelin, ''Ancient Botany'', Routledge, 2015, p. 8. was the successor to Aristotle in the Peripatetic ...

attempted to improve. There are also passages in Aristotle's work, such as the famous sea-battle argument in ''De Interpretatione
''De Interpretatione'' or ''On Interpretation'' (Ancient Greek, Greek: Περὶ Ἑρμηνείας, ''Peri Hermeneias'') is the second text from Aristotle's ''Organon'' and is among the earliest surviving philosophical works in the Western phil ...

'' §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#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a countr ...

, Philo the Dialectician
Philo the Dialectician ( el, Φίλων; fl.
''Floruit'' (), abbreviated fl. (or occasionally flor.), Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communic ...

and the Stoic Chrysippus
Chrysippus of Soli (; grc-gre, Χρύσιππος ὁ Σολεύς, ; ) was a Greek Stoic philosopher
A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosop ...

each developed a modal system that accounted for the interdefinability of possibility and necessity, accepted axiom
An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

T (see below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Fred Below (1926–1988), American blues drummer
*Fritz von Below (1853 ...

), and combined elements of modal logic and temporal logic in attempts to solve the notorious Diodorus Cronus#Master Argument, Master Argument. The earliest formal system of modal logic was developed by Avicenna, who ultimately developed a theory of "Temporal logic, temporally modal" syllogistic.History of logic: Arabic logic''Encyclopædia Britannica''. Modal logic as a self-aware subject owes much to the writings of the Scholastics, in particular William of Ockham and John Duns Scotus, who reasoned informally in a modal manner, mainly to analyze statements about essence and accident (philosophy), accident. In the 19th century, Hugh MacColl made innovative contributions to modal logic, but did not find much acknowledgment. C. I. Lewis 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 Vacuous truth, a falsehood implies any proposition. This work culminated in his 1932 book ''Symbolic Logic'' (with Cooper Harold Langford, C. H. Langford), which introduced the five systems ''S1'' through ''S5''. After Lewis, modal logic received little attention for several decades. Nicholas Rescher has argued that this was because Bertrand Russell 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 functions," as he wrote in ''The Analysis of Matter''. Arthur Norman Prior warned Ruth Barcan Marcus to prepare well in the debates concerning quantified modal logic with Willard Van Orman Quine, due to the biases against modal logic. Ruth C. Barcan (later Ruth Barcan Marcus) 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

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 Kripke semantics, relational semantics for modal logic. In relational semantics, a modal formula's truth value at a ''possible world'' w ...

* Conceptual necessity
* Counterpart theory
* David Kellogg Lewis
* De dicto and de re, ''De dicto'' and ''de re''
* Description logic
* Doxastic logic
* Dynamic logic (modal logic), Dynamic logic
* Enthymeme
* Free choice inference
* Hybrid logic
* Interior algebra
* Interpretability logic
* Notes

References

* ''This article includes material from the'' Free On-line Dictionary of Computing, ''used with Wikipedia:Foldoc license, permission under the'' GFDL. * 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.; Johan van Benthem (logician), 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. * Max Cresswell, 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 analytic tableau, 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. * Bjarni Jónsson, Jónsson, B. and Alfred Tarski, 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. * John Lemmon, Lemmon, E. J. (with Dana Scott, Scott, D.) (1977) ''An Introduction to Modal Logic'', American Philosophical Quarterly Monograph Series, no. 11 (Krister Segerberg, series ed.). Basil Blackwell. * Clarence Irving Lewis, Lewis, C. I. (with Cooper Harold Langford, Langford, C. H.) (1932). ''Symbolic Logic''. Dover reprint, 1959. * Arthur Prior, 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.

"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, . [Covers 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 (computer scientist), 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 topology, topological semantics for the modal logic S4.

LoTREC

The most generic prover for modal logics from IRIT/Toulouse University {{Authority control Logic Modal logic, Philosophical logic Mathematical logic Semantics