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| [[File:DavidLewis2.jpg|thumb|David Lewis]]
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| {{Refimprove|date=August 2008}}
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| '''Modal logic''' is a type of [[mathematical logic#Formal logic|formal logic]] primarily developed in the 1960s that extends classical [[Propositional logic|propositional]] and [[predicate logic]] to include operators expressing [[Linguistic modality|modality]]. Modals—words that express modalities—qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is [[usually]] happy, in which case the term "usually" is functioning as a modal. The traditional [[Alethic modality|alethic modalities]], or modalities of truth, include [[Logical possibility|possibility]] ("Possibly, p", "It is possible that p"), necessity ("Necessarily, p", "It is necessary that p"), and impossibility ("Impossibly, p", "It is impossible that p").<ref>"Formal Logic", by A. N. Prior, Oxford Univ. Press, 1962, p. 185</ref> Other modalities that have been formalized in modal logic include [[Temporal Logic|temporal]] modalities, or modalities of time (notably, "It was the case that p", "It has always been that p", "It will be that p", "It will always be that p"),<ref>"Temporal Logic", by Rescher and Urquhart, Springer-Verlag, 1971, p. 52</ref><ref>"Past, Present and Future", by A. N. Prior, Oxford Univ. Press, 1967</ref> [[Deontic logic|deontic]] modalities (notably, "It is obligatory that p", and "It is permissible that p"), [[Epistemic logic|epistemic]] modalities, or modalities of knowledge ("It is known that p")<ref>"Knowledge and Belief", by Jaakko Hinntikka, Cornell Univ. Press, 1962</ref> and [[Doxastic logic|doxastic]] modalities, or modalities of belief ("It is believed that p").<ref>"Topics in Philosophical Logic", by N. Rescher, Humanities Press, 1968, p. 41</ref>
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| A formal modal logic represents modalities using [[modal operator]]s. For example, "It might rain today" and "It is possible that rain will fall today" both contain the notion of possibility. In a modal logic this is represented as an operator, Possibly, attached to the sentence "It will rain today".
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| The basic [[Unary operation|unary]] (1-place) modal operators are usually written □ for Necessarily and ◇ for Possibly. In a [[classical modal logic]], each can be expressed by the other with [[negation]]:
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| :<math>\Diamond P \leftrightarrow \lnot \Box \lnot P; \;\!</math>
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| :<math>\Box P \leftrightarrow \lnot \Diamond \lnot P. \;\!</math>
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| Thus it is ''possible'' that it will rain today if and only if it is ''not necessary'' that it will ''not'' rain today; and it is ''necessary'' that it will rain today if and only if it is ''not possible'' that it will ''not'' rain today. Alternative symbols used for the modal operators are "L" for Necessarily and "M" for Possibly.<ref>So in the standard work ''A New Introduction to Modal Logic'', by G. E. Hughes and M. J. Cresswell, Routledge, 1996, ''passim''.</ref>
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| ==Development of modal logic==
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| In addition to his non-modal syllogistic, [[Aristotle]] also developed a modal syllogistic in Book I of his ''[[Prior Analytics]]'' (chs 8-22), which [[Theophrastus]] attempted to improve.<ref>{{Sep entry|logic-ancient|Ancient Logic| Susanne Bobzien}}</ref> There are also passages in Aristotle's work, such as the famous [[problem of future contingents|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]], [[Philo the Dialectician]] and the Stoic [[Chrysippus]] each developed a modal system that accounted for the interdefinability of possibility and necessity, accepted axiom T and combined elements of modal logic and [[temporal logic]] in attempts to solve the notorious Master Argument.<ref>Bobzien, S. (1993). "Chrysippus' Modal Logic and its Relation to Philo and Diodorus", in K. Doering & Th. Ebert (eds), ''Dialektiker und Stoiker'', Stuttgart 1993, pp. 63-84.</ref> The earliest formal system of modal logic was developed by [[Avicenna]], who ultimately developed a theory of "[[Temporal logic|temporally]] modal" syllogistic.<ref name=Britannica>[http://www.britannica.com/ebc/article-65928 History of logic: Arabic logic], ''[[Encyclopædia Britannica]]''.</ref> 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]].
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| [[C. I. Lewis]] founded modern modal logic in his 1910 Harvard thesis and in a series of scholarly articles beginning in 1912. This work culminated in his 1932 book ''Symbolic Logic'' (with [[Cooper Harold Langford|C. H. Langford]]), which introduced the five systems ''S1'' through ''S5''.
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| Ruth C. Barcan (later [[Ruth Barcan Marcus]]) developed the first axiomatic systems of quantified modal logic — first and second order extensions of Lewis's "S2", "S4", and "S5".
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| The contemporary era in modal semantics began in 1959, when [[Saul Kripke]] (then only a 19-year-old [[Harvard University]] undergraduate) introduced the now-standard [[Kripke semantics]] 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]].
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| [[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), [[computational tree logic]] (CTL), [[Hennessy-Milner logic|Hennessy–Milner logic]], and ''T''.
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| The mathematical structure of modal logic, namely [[Boolean algebra (structure)|Boolean algebra]]s augmented with [[unary operation]]s (often called [[modal algebra]]s), began to emerge with [[J.C.C. McKinsey|J. C. C. McKinsey]]'s 1941 proof that ''S2'' and ''S4'' are decidable,<ref>{{cite journal|author=McKinsey, J. C. C.|title=A Solution of the Decision Problem for the Lewis Systems S2 and S4, with an Application to Topology|journal=J. Symb. Logic|year=1941|volume=6|issue=4|pages=117–134|jstor=2267105}}</ref> and reached full flower in the work of [[Alfred Tarski]] and his student [[Bjarni Jonsson]] (Jonsson 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 operator]]s of [[topology]]. Texts on modal logic typically do little more than mention its connections with the study of [[Boolean algebra (structure)|Boolean algebra]]s and [[topology]]. For a thorough survey of the history of formal modal logic and of the associated mathematics, see [[Robert Goldblatt]] (2006).<ref>[http://www.mcs.vuw.ac.nz/~rob/papers/modalhist.pdf ]</ref>
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| ==Formalizations==
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| ===Semantics===
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| The semantics for modal logic are usually given as follows:<ref>Fitting and Mendelsohn. ''First-Order Modal Logic''. Kluwer Academic Publishers, 1998. Section 1.6</ref>
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| First we define a ''frame'', which consists of a non-empty set, ''G'', whose members are generally called possible worlds, and a binary relation, ''R'', that holds (or not) between the possible worlds of ''G''. This binary relation is called the ''[[accessibility relation]]''. For example, ''w'' ''R'' ''v'' means that the world ''v'' is accessible from world ''w''. That is to say, the state of affairs known as ''v'' is a live possibility for ''w''. This gives a pair, <math>\langle G, R\rangle</math>.
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| Next, the ''frame'' is extended to a ''model'' by specifying the [[truth-value]]s of all propositions at each of the worlds in ''G''. We do so by defining a relation ''v'' between possible worlds and positive literals. If there is a world ''w'' such that <math>v(w, P)</math>, then ''P'' is true at ''w''. A model is thus an ordered triple, <math>\langle G, R, v \rangle</math>.
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| Then we recursively define the truth of a formula in a model:
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| * if <math>v(w, P)</math> then <math>w \models P</math>
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| * <math>w \models \neg P</math> if and only if <math>w \not \models P</math>
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| * <math>w \models (P \wedge Q) </math> if and only if <math>w \models P</math> and <math>w \models Q</math>
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| * <math>w \models \Box P</math> if and only if for every element ''u'' of ''G'', if ''w'' ''R'' ''u'' then <math>u \models P</math>
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| * <math>w \models \Diamond P</math> if and only if for some element ''u'' of ''G'', it holds that ''w'' ''R'' ''u'' and <math>u \models P</math>
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| According to these semantics, a truth is ''necessary'' with respect to a possible world ''w'' if it is true at every world that is accessible to ''w'', and ''possible'' if it is true at some world that is accessible to ''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.
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| It should also be noted that the definition of <math>\Box</math> makes vacuously true certain sentences, since when it speaks of "every world that is accessible to w" it takes for granted the usual mathematical interpretation of the word "every" (see [[vacuous truth]]). Hence, if a world w doesn't have any accessible worlds, any sentence beginning with <math>\Box</math> is true.
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| The different systems of modal logic are distinguished by the properties of their corresponding accessibility relations. There are several systems that have been espoused (often called ''frame conditions''). An accessibility relation is:
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| * '''[[reflexive relation|reflexive]]''' iff ''w'' ''R'' ''w'', for every ''w'' in ''G''
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| * '''[[symmetric relation|symmetric]]''' iff ''w'' ''R'' ''u'' implies ''u'' ''R'' ''w'', for all ''w'' and ''u'' in ''G''
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| * '''[[transitive relation|transitive]]''' iff ''w'' ''R'' ''u'' and ''u'' ''R'' ''q'' together imply ''w'' ''R'' ''q'', for all ''w'', ''u'', ''q'' in ''G''.
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| * '''[[serial relation|serial]]''' iff, for each ''w'' in ''G'' there is some ''u'' in ''G'' such that ''w'' ''R'' ''u''.
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| * '''[[euclidean relation|euclidean]]''' iff, for every ''u'', ''t'', and ''w'', ''w'' ''R'' ''u'' and ''w'' ''R'' ''t'' implies ''u'' ''R'' ''t'' (note that it also implies: ''t'' ''R'' ''u'')
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| The logics that stem from these frame conditions are:
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| *'''K''' := no conditions
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| *'''D''' := serial
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| *'''T''' := reflexive
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| *'''S4''' := [[preorder|reflexive and transitive]]
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| *'''S5''' := reflexive, symmetric, transitive and euclidean
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| S5 models are reflexive, transitive, and euclidean; this implies that for S5 models the accessibility relation ''R'' is an equivalence relation; the relation ''R'' is reflexive, symmetric and transitive. It is interesting to note how the euclidean property along with reflexivity yields symmetry and transitivity. (The euclidean property can be obtained, as well, from symmetry and transitivity.) We can prove that these frames produce the same set of valid sentences as do any 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).
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| For example, in S4:
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| :: <math>w \models \Diamond P</math> if and only if for some element ''u'' of ''G'', it holds that <math>u \models P</math> and ''w'' ''R'' ''u''.
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| However, in S5, we can just say that
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| :: <math>w \models \Diamond P</math> if and only if for some element ''u'' of ''G'', it holds that <math>u \models P</math>.
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| We can drop the accessibility clause from the latter stipulation because it is trivially true of all S5 frames that ''w'' ''R'' ''u''.
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| All of these logical systems can also be defined axiomatically, as is shown in the next section. For example, in S5, the axioms P → <math>\Box</math><math>\Diamond</math>P, <math>\Box</math>P → <math>\Box</math><math>\Box</math>P, and <math>\Box</math>P → 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.
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| ===Axiomatic systems===
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| The first formalizations of modal logic were axiomatic. Numerous variations with very different properties have been proposed since [[C. I. Lewis]] began working in the area in 1910. Hughes and Cresswell (1996), for example, describe 42 normal and 25 non-normal modal logics. Zeman (1973) describes some systems Hughes and Cresswell omit.
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| Modern treatments of modal logic begin by augmenting the [[propositional calculus]] with two unary operations, one denoting "necessity" and the other "possibility". The notation of [[Clarence Irving Lewis|C. I. Lewis]], much employed since, denotes "necessarily ''p''" by a prefixed "box" (<!--'''[]p'''--> <math>\Box p </math>) whose scope is established by parentheses. Likewise, a prefixed "diamond" (<!--'''<>p'''--><math>\Diamond p</math>) denotes "possibly ''p''". Regardless of notation, each of these operators is definable in terms of the other:
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| * <!--[]p--><math>\Box p</math> (necessarily ''p'') is equivalent to <!-- ~<>~p --><math>\neg \Diamond \neg p </math> ("not possible that not-''p''")
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| * <!-- <>p --> <math>\Diamond p </math> (possibly ''p'') is equivalent to <!-- ~[]~p --><math>\neg \Box \neg p </math> ("not necessarily not-''p''")
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| Hence <math>\Box</math> and <math>\Diamond</math> form a [[duality (mathematics)#Duality in logic and set theory|dual pair]] of operators.
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| In many modal logics, the necessity and possibility operators satisfy the following analogs of [[de Morgan's laws]] from [[Boolean algebra (logic)|Boolean algebra]]:
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| :"It is '''not necessary that''' ''X''" is [[Logical equivalence|logically equivalent]] to "It is '''possible that not''' ''X''".
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| :"It is '''not possible that''' ''X''" is logically equivalent to "It is '''necessary that not''' ''X''".
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| Precisely what axioms and rules must be added to the [[propositional calculus]] 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]]s, include the following rule and axiom:
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| * '''N''', '''Necessitation Rule''': If ''p'' is a [[theorem]] (of any system invoking '''N'''), then <math>\Box p</math> is likewise a theorem.
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| * '''K''', '''Distribution Axiom''': <!--[](''p'' → ''q'') → ([]p → []q)--><math> \Box (p \rightarrow q) \rightarrow (\Box p \rightarrow \Box q)</math>.
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| The weakest [[normal modal logic]], named ''K'' in honor of [[Saul Kripke]], is simply the [[propositional calculus]] augmented by <math>\Box</math>, 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 --> <math> \Box p </math> is true then <!-- [][]p --> <math> \Box \Box p </math> 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.
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| 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:
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| *'''T''', '''Reflexivity Axiom''': <!-- []p → p --><math> \Box p \rightarrow p </math> (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 ''S1<sup>0</sup>''.
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| Other well-known elementary axioms are:
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| *'''4''': <math> \Box p \rightarrow \Box \Box p</math>
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| *'''B''': <math> p \rightarrow \Box \Diamond p</math>
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| *'''D''': <math> \Box p \rightarrow \Diamond p</math>
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| *'''5''': <math> \Diamond p \rightarrow \Box \Diamond p </math>
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| These yield the systems (axioms in bold, systems in italics):
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| *''K'' := '''K''' + '''N'''
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| *''T'' := ''K'' + '''T'''
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| *''S4'' := ''T'' + '''4'''
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| *''S5'' := ''S4'' + '''5'''
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| *''D'' := ''K'' + '''D'''.
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| ''K'' through ''S5'' form a nested hierarchy of systems, making up the core of [[normal modal logic]]. But specific rules or sets of rules may be appropriate for specific systems. For example, in deontic logic, <math> \Box p \rightarrow \Diamond p</math> (If it ought to be that ''p'', then it is permitted that ''p'') seems appropriate, but we should probably not include that <math> p \rightarrow \Box \Diamond p</math>. In fact, to do so is to commit the [[naturalistic 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).
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| 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.
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| ==Alethic logic==
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| {{main|Alethic modality}}
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| Modalities of necessity and possibility are called ''alethic'' modalities. They are also sometimes called ''special'' modalities, from the [[Latin]] ''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, and so on, than it is to make sense of relativizing other notions.
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| In [[classical modal logic]], a proposition is said to be
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| *'''possible''' if and only if it is ''not necessarily false'' (regardless of whether it is actually true or actually false);
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| *'''necessary''' if and only if it is ''not possibly false''; and
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| *'''contingent''' if and only if it is ''not necessarily false'' and ''not necessarily true'' (i.e. possible but not necessarily true).
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| In classical modal logic, therefore, either the notion of 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]]. [[Intuitionistic modal logic]] treats possibility and necessity as not perfectly symmetric.
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| 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 [[Gottfried Wilhelm Leibniz|Leibniz]]) 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]].
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| ===Physical possibility===
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| Something is physically, or nomically, possible if it is permitted by the [[physical law|laws of physics]].<sup>[source?]</sup> For example, current theory is thought to allow for there to be an [[atom]] with an [[atomic number]] of 126,<ref>{{cite web|title=Superheavy Element 114 Confirmed: A Stepping Stone to the Island of Stability|url=http://phys.org/news173028810.html|publisher=phys.org}}</ref> even if there are no such atoms in existence. In contrast, while it is logically possible to accelerate beyond the [[speed of light]],<ref name="Feinberg67">
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| {{cite journal
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| |last=Feinberg |first=G.
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| |year=1967
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| |title=Possibility of Faster-Than-Light Particles
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| |journal=[[Physical Review]]
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| |volume=159 |issue=5 |pages=1089–1105
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| |bibcode=1967PhRv..159.1089F
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| |doi=10.1103/PhysRev.159.1089
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| }} See also Feinberg's later paper: Phys. Rev. D 17, 1651 (1978)</ref> modern science stipulates that it is not physically possible for material particles or information.<ref>{{cite journal | last = Einstein | first = Albert | authorlink = Albert Einstein | title = Zur Elektrodynamik bewegter Körper | journal = Annalen der Physik | volume = 17 | pages = 891–921 | date = 1905-06-30|bibcode = 1905AnP...322..891E |doi = 10.1002/andp.19053221004 | issue = 10 }}</ref>
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| ===Metaphysical possibility===
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| [[Philosophers]]{{who|date=April 2012}} ponder the properties that objects have independently of those dictated by scientific laws. For example, it might be metaphysically necessary, as some have thought, that all thinking beings have bodies and can experience the passage of [[time]].{{citation needed|date=April 2012}} [[Saul Kripke]] has argued that every person necessarily has the parents they do have: anyone with different parents would not be the same person.<ref>Saul Kripke. ''Naming and Necessity''. Harvard University Press, 1980. pg 113</ref>
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| Metaphysical possibility is generally thought{{by whom|date=April 2012}} to be more restricting than bare logical possibility (i.e., fewer things are metaphysically possible than are logically possible). Its exact relation to physical possibility is a matter of some dispute. Philosophers{{who|date=April 2012}} 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.
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| ===Confusion with epistemic modalities===
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| Alethic modalities and epistemic modalities (see below) are often expressed in English using the same words. "It is possible that bigfoot exists" can mean either "Bigfoot ''could'' exist, whether or not bigfoot does in fact exist" (alethic), or more likely, "For all I know, bigfoot exists" (epistemic).
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| It has been questioned whether these 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).<ref>{{cite book|last=Eschenroeder|first=Erin|coauthors=Sarah Mills; Thao Nguyen|title=The Expression of Modality|editor=William Frawley|publisher=Mouton de Gruyter|date=2006-09-30|series=The Expression of Cognitive Categories|pages=8–9|url=http://books.google.co.uk/books?id=72URszHq2SEC&pg=PT18|isbn=3-11-018436-2|accessdate=2010-01-03}}</ref> 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]].<ref>{{cite book|last=Nuyts|first=Jan|title=Epimestic Modality, Language, and Conceptualization: A Cognitive-pragmatic Perspective|publisher=John Benjamins Publishing Co|date=November 2000|series=Human Cognitive Processing|page=28|isbn=90-272-2357-2}}</ref>
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| ==Epistemic logic==
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| {{Main|Epistemic logic}}
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| '''Epistemic modalities''' (from the Greek ''episteme'', knowledge), deal with the ''certainty'' of sentences. The <math> \Box </math> operator is translated as "x knows that…", and the <math> \Diamond </math> 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:
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| 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, Bigfoot possibly ''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'' (which is not equivalent to "it is ''possible that'' Bigfoot exists – for all I know", which contradicts (1)).
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| 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.<ref>See [[Goldbach's_conjecture#Origins|Goldbach's conjecture - Origins]]</ref>
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| 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.
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| 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 <math>\Box P \rightarrow \Box \Box P</math> 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 [[Modal_logic#Axiomatic_systems|the section on axiomatic systems]]):
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| * '''K''', ''Distribution Axiom'': <!--[](''p'' → ''q'') → ([]p → []q)--><math> \Box (p \rightarrow q) \rightarrow (\Box p \rightarrow \Box q)</math>.
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| Some schools of thought have found this to be disconcerting, noting that with '''K''', we can prove that we know all the [[logical consequence]]s of our beliefs: If ''q'' is a logical consequence of ''p'', then <math> \Box (p \rightarrow q)</math>. And if so, then we can deduce that <math>(\Box p \rightarrow \Box q)</math> using '''K'''. When we translate this into epistemic terms, this says that if ''q'' is a logical consequence of ''p'', then we know that it is, and if we know ''p'', we know ''q''. That is to say, we know all the logical consequences of our beliefs. This must be true for all possible [[Kripke semantics|Kripkean]] modal interpretations of epistemic cases where <math>\Box</math> is translated as "knows that". But then, for example, if ''x'' knows that prime numbers are divisible only by themselves and the number one, then ''x'' knows that 2<sup>31</sup> − 1 is prime (since this number is only divisible by itself and the number one). That is to say, under the modal interpretation of knowledge, anyone who knows the definition of a prime number knows that this number is prime. This shows that epistemic modal logics that are based on normal modal systems provide an idealized account of knowledge, and explain objective, rather than subjective knowledge (if anything).
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| However, the deduction that K implies that we know all the logical consequences of our beliefs is heavily dependent on semantics (much to the word "all") and may be faulty. We can read the distribution axiom as, "If it is necessary that p implies q, then necessarily p implies that q is necessary." which does not imply that p or q are true in either case, but just shows a relationship (if, then). However, including "x knows" in the statement might make it seem to lose some of that ambiguity. "If x knows that 'p implies q', then 'x knows that p' implies 'x knows that q'.". This can be further rewritten as, "x knows that 'p implies q', implies, 'x knows that p' implies 'x knows that q'.". To make this more readable, we can say, "if x knows that p implies q, then whenever x knows that p, x also knows that q.". The prime number example above is faulty. A correct application of the Distribution Axiom K would be worded: "If x knows that prime numbers are divisible only by themselves and the number one, then whenever x knows that a number is prime, x also knows that the number is divisible only by itself and the number 1". That is, "if x knows that 2<sup>31</sup> − 1 is divisible only by itself and the number 1, then x knows that 2<sup>31</sup> − 1 is a prime number.". Those statements can NOT be used to say that "If x knows that a prime number is divisible by itself and the number 1, then x knows that 2<sup>31</sup> − 1 is a prime number." Because x just simply doesn't have enough information to draw that conclusion. Therefore, for all x knows, K might still be disconcerting.
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| ==Temporal logic==
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| {{Main|Temporal logic}}
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| Temporal logic is an approach to the semantics of expressions with [[Grammatical tense|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.
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| 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:
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| :'''F'''''P'' : It will sometimes be the case that ''P''
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| :'''G'''''P'' : It will always be the case that ''P''
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| :'''P'''''P'' : It was sometime the case that ''P''
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| :'''H'''''P'' : It has always been the case that ''P''
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| There are then at least three modal logics that we can develop. For example, we can stipulate that,
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| :<math> \Diamond P </math> = ''P'' is the case at some time ''t''
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| :<math> \Box P </math> = ''P'' is the case at every time ''t''
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| Or we can trade these operators to deal only with the future (or past). For example,
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| :<math> \Diamond_1 P </math> = '''F'''''P''
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| :<math> \Box_1 P </math> = '''G'''''P''
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| or,
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| :<math> \Diamond_2 P</math> = ''P'' and/or '''F'''''P''
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| :<math> \Box_2 P </math> = ''P'' and '''G'''''P''
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| The operators '''F''' and '''G''' may seem initially foreign, but they create [[normal modal logic|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'' → <math> \Box </math>'''P'''''P'' says (effectively), ''Everything that is past and true is necessary''.
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| 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.
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| Similarly, the [[problem of future contingents]] 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]] to reject the [[principle of bivalence]] for assertions concerning the future.
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| Additional binary operators are also relevant to temporal logics, ''q.v.'' [[Linear Temporal Logic]].
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| Versions of temporal logic can be used in [[computer science]] to model computer operations and prove theorems about them. In one version, <math>\Diamond P</math> means "at a future time in the computation it is possible that the computer state will be such that P is true"; <math>\Box P</math> means "at all future times in the computation P will be true". In another version, <math>\Diamond P</math> means "at the immediate next state of the computation, P might be true"; <math>\Box P</math> means "at the immediate next state of the computation, P will be true". These differ in the choice of [[Accessibility relation]]. (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.
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| ==Deontic logic==
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| {{Main|Deontic logic}}
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| Likewise talk of morality, or of [[obligation]] and [[norm (philosophy)|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 logic|deontic]]'', from the Greek for "duty".
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| Deontic logics commonly lack the axiom '''T''' semantically corresponding to the reflexivity of the accessibility relation in [[Kripke semantics]]: in symbols, <math>\Box\phi\to\phi</math>. Interpreting <math>\Box</math> 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.
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| Instead, using [[Kripke semantics]], 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.<ref>See, e.g., {{cite journal |first=Sven |last=Hansson |title=Ideal Worlds—Wishful Thinking in Deontic Logic |journal=Studia Logica |volume=82 |issue=3 |pages=329–336 |year=2006 |doi=10.1007/s11225-006-8100-3 }}</ref>
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| One other principle that is often (at least traditionally) accepted as a deontic principle is ''D'', <math>\Box\phi\to\Diamond\phi</math>, 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.)
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| ===Intuitive problems with deontic logic===
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| When we try and 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,
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| : (1) <math>(K \rightarrow \Box Q)</math>
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| : (2) <math>\Box (K \rightarrow Q)</math>
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| But (1) and ''K'' together entail <math>\Box Q</math>, 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 <math>\Box (K \rightarrow (K \and \lnot Q))</math>. Now suppose (as seems reasonable) that you ought not to steal anything, or <math>\Box \lnot K</math>. But then we can deduce <math>\Box (K \rightarrow (K \and \lnot Q))</math> via <math>\Box (\lnot K) \rightarrow \Box (K \rightarrow K \and \lnot K)</math> and <math>\Box (K \and \lnot K \rightarrow (K \and \lnot Q)) </math> (the [[contrapositive]] of <math>Q \rightarrow K</math>); 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.<ref>Ted Sider's ''Logic for Philosophy'', unknown page. <!-- http://homepages.nyu.edu/~ts65/books/lfp/lfp.html Link is dead as of Dec 20, 2010 --></ref>
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| == Doxastic logic ==
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| {{Main|Doxastic logic}}
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| ''Doxastic logic'' concerns the logic of belief (of some set of agents). The term doxastic is derived from the [[ancient Greek]] ''doxa'' which means "belief". Typically, a doxastic logic uses <math>\Box</math>, often written "B", to mean "It is believed that", or when relativized to a particular agent s, "It is believed by s that".
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| ==Other modal logics==
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| {{See also|Intensional logic}}
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| Significantly, modal logics can be developed to accommodate most of these idioms; it is the fact of their common logical structure (the use of "intensional" sentential operators) that make them all varieties of the same thing.
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| == The ontology of possibility ==
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| {{Further| Accessibility relation|Possible worlds}}
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| In the most common interpretation of modal logic, one considers "[[logically possible]] worlds". If a statement is true in all [[possible worlds]], 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.
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| 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]] 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.<ref>Kripke, Saul. ''Naming and Necessity''. (1980; Harvard UP), pp. 43–5.</ref> 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.<ref>Kripke, Saul. ''Naming and Necessity''. (1980; Harvard UP), pp. 15–6.</ref> [[David Lewis (philosopher)|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 – ''[[Indexicality|this]]'' world.<ref>David Lewis, ''On the Plurality of Worlds'' (1986; Blackwell)</ref> 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 Merrihew Adams|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.<ref>Adams, Robert M. [http://www.jstor.org/stable/2214751 ''Theories of Actuality'']. Noûs, Vol. 8, No. 3 (Sep., 1974), particularly pp. 225–31.</ref>
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| 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".
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| ==Further applications==
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| Modal logics have begun to be used in areas of the humanities such as literature, poetry, art and history.<ref>See http://www.estherlederberg.com/EImages/Extracurricular/Dickens%20Universe/Counter%20Factuals.html</ref><ref>Andrew H. Miller, "Lives Unled in Realist Fiction", Representations 98, Spring 2007, The Regents of the University of California, ISSN 1553-855X, pp. 118-134</ref><ref>See also http://www.estherlederberg.com/EImages/Extracurricular/Dickens%20Universe/Page%2017%20CounterFactuals.html</ref>
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| ==Controversies==
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| [[Nicholas Rescher]] has argued that [[Bertrand Russell]] rejected Modal Logic, and that this rejection led to the theory of modal logic languishing for decades.<ref>{{cite book|last=Rescher|first=Nicholas|title=Bertrand Russell Memorial Volume|year=1979|publisher=George Allen and Unwin|location=London|pages=146|editor=George W. Roberts|chapter=Russell and Modal Logic}}</ref> 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]]s," as he wrote in ''[[The Analysis of Matter]]''.<ref>{{cite journal|last=Dejnozka|first=Jan|title=Ontological Foundations of Russell's Theory of Modality|journal=Erkenntnis|year=1990|volume=32|pages=383–418|url=http://www.members.tripod.com/~Jan_Dejnozka/onto_found_russell_modality.pdf|accessdate=2012-10-22}}; quote is cited from {{cite book|last=Russell|first=Bertrand|title=The Analysis of Matter|year=1927|pages=173}}</ref>
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| [[Arthur Norman Prior]] warned his protégé [[Ruth Barcan]] to prepare well in the debates concerning Quantified Modal Logic with [[Willard Van Orman Quine]], due to the biases against Modal Logic.<ref>"Modalities: Philosophical Essays", by Ruth Barcan Marcus, Oxford Univ. Press, 1993, Chapter 14</ref>
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| ==See also==
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| {{Col-begin}}
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| {{Col-1-of-4}}
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| *[[Accessibility relation]]
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| *[[Counterpart theory]]
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| *[[David Kellogg Lewis]]
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| *[[De dicto and de re]]
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| *[[Description logic]]
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| *[[Doxastic logic]]
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| *[[Dynamic logic (modal logic)|Dynamic logic]]
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| {{Col-2-of-4}}
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| *[[Enthymeme]]
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| *[[Hybrid logic]]
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| *[[Interior algebra]]
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| *[[Interpretability logic]]
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| *[[Kripke semantics]]
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| *[[Modal verb]]
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| *[[Multi-valued logic]]
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| {{Col-3-of-4}}
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| *[[Physics envy]]
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| *[[Possible worlds]]
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| *[[Provability logic]]
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| *[[Regular modal logic]]
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| *[[Relevance logic]]
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| *[[Research Materials: Max Planck Society Archive]]
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| *[[Rhetoric]]
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| *[[Strict conditional]]
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| *[[Two dimensionalism]]
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| {{Col-4-of-4}}
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| {{Portal|Logic|Thinking}}
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| {{Col-end}}
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| ==Notes==
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| {{Reflist|30em}}
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| ==References==
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| *''This article includes material from the'' [[Free On-line Dictionary of Computing]], ''used with [[Wikipedia:Foldoc license|permission]] under the'' [[GFDL]].
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| *Blackburn, Patrick; de Rijke, Maarten; and Venema, Yde (2001) ''Modal Logic''. Cambridge University Press. ISBN 0-521-80200-8
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| *Barcan-Marcus, Ruth JSL 11 (1946) and JSL 112 (1947) and "Modalities", OUP, 1993, 1995.
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| *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).
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| *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).
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| *Blackburn, P.; [[Johan van Benthem (logician)|van Benthem, J.]]; and Wolter, Frank; Eds. (2006) ''[http://www.csc.liv.ac.uk/~frank/MLHandbook/ Handbook of Modal Logic]''. North Holland.
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| *Chagrov, Aleksandr; and Zakharyaschev, Michael (1997) ''Modal Logic''. Oxford University Press. ISBN 0-19-853779-4
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| *Chellas, B. F. (1980) ''Modal Logic: An Introduction''. Cambridge University Press. ISBN 0-521-22476-4
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| *[[Max Cresswell|Cresswell, M. J.]] (2001) "Modal Logic" in Goble, Lou; Ed., ''The Blackwell Guide to Philosophical Logic''. Basil Blackwell: 136–58. ISBN 0-631-20693-0
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| *Fitting, Melvin; and Mendelsohn, R. L. (1998) ''First Order Modal Logic''. Kluwer. ISBN 0-7923-5335-8
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| *[[James Garson]] (2006) ''Modal Logic for Philosophers''. Cambridge University Press. ISBN 0-521-68229-0. A thorough introduction to modal logic, with coverage of various derivation systems and a distinctive approach to the use of diagrams in aiding comprehension.
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| *Girle, Rod (2000) ''Modal Logics and Philosophy''. Acumen (UK). ISBN 0-7735-2139-9. Proof by [[analytic tableau|refutation trees]]. A good introduction to the varied interpretations of modal logic.
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| *[http://www.mcs.vuw.ac.nz/~rob/ Goldblatt, Robert] (1992) "Logics of Time and Computation", 2nd ed., CSLI Lecture Notes No. 7. University of Chicago Press.
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| *—— (1993) ''Mathematics of Modality'', CSLI Lecture Notes No. 43. University of Chicago Press.
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| *—— (2006) "[http://www.mcs.vuw.ac.nz/~rob/papers/modalhist.pdf 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.
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| *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.
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| *Hughes, G. E., and Cresswell, M. J. (1996) ''A New Introduction to Modal Logic''. Routledge. ISBN 0-415-12599-5
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| *[[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.
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| *Kracht, Marcus (1999) ''Tools and Techniques in Modal Logic'', Studies in Logic and the Foundations of Mathematics No. 142. North Holland.
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| *[[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.
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| *[[Clarence Irving Lewis|Lewis, C. I.]] (with [[Cooper Harold Langford|Langford, C. H.]]) (1932). ''Symbolic Logic''. Dover reprint, 1959.
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| *[[Arthur Prior|Prior, A. N.]] (1957) ''Time and Modality''. Oxford University Press.
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| *Snyder, D. Paul "Modal Logic and its applications", Van Nostrand Reinhold Company, 1971 (proof tree methods).
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| *Zeman, J. J. (1973) ''[http://www.clas.ufl.edu/users/jzeman/modallogic/ Modal Logic.]'' Reidel. Employs [[Polish notation]].
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| *History of logic, Encyclopædia Britannica.
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| ==Further reading==
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| *[[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, ISBN 0-444-50826-0. 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.
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| ==External links==
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| *[[Stanford Encyclopedia of Philosophy]]:
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| **"[http://plato.stanford.edu/entries/logic-modal Modal logic]" – by [[James Garson]].
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| **"[http://plato.stanford.edu/entries/logic-provability/ Provability Logic]" – by Rineke Verbrugge.
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| *[[Edward N. Zalta]], 1995, "[http://mally.stanford.edu/notes.pdf Basic Concepts in Modal Logic.]"
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| *[[John McCarthy (computer scientist)|John McCarthy]], 1996, "[http://www-formal.stanford.edu/jmc/mcchay69/node22.html Modal Logic.]"
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| *[http://molle.sourceforge.net/ Molle] a Java prover for experimenting with modal logics
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| *Suber, Peter, 2002, "[http://www.earlham.edu/~peters/courses/logsys/nonstbib.htm#modal Bibliography of Modal Logic.]"
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| *Marcus, Ruth Barcan, "Modalities" OUP 1993, 1995
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| *[http://www.cc.utah.edu/~nahaj/logic/structures/systems/index.html List of Logic Systems] List of many modal logics with sources, by John Halleck.
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| *[http://aiml.net/ Advances in Modal Logic.] Biannual international conference and book series in modal logic.
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| *[http://teachinglogic.imag.fr/TableauxS4 S4prover] A tableaux prover for S4 logic
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| *"[http://www.labri.fr/perso/moot/talks/TopologyNotes.pdf Some Remarks on Logic and Topology]" – by Richard Moot; exposits a [[topology|topological]] [[semantics]] for the modal logic S4.
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| {{Logic}}
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| {{DEFAULTSORT:Modal Logic}}
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| [[Category:Non-classical logic]]
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| [[Category:Modal logic| ]]
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| [[Category:Philosophical logic]]
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| [[Category:Modality]]
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