Modal logic
In formal logic, a modal logic is any logic for handling modalities: concepts like possibility, existence, and necessity. Logics for handling a number of other ideas, such as eventually, formerly, can, could, might, may, must are by extension also called modal logics, since it turns out that these can be treated in similar ways.
A formal modal logic represents modalities using unary modal operators. For example, "Jones's murder was a possibility"; "Jones was possibly murdered"; and "It is possible that Jones was murdered," all contain the notion of possibility; in a modal logic this is represented as an operator, Possibly, attaching to the sentence Jones was murdered.
The basic modal operators are usually written \Box (or L) for Necessarily and \Diamond (or M) for Possibly. In a classical modal logic, each can be expressed by the other and negation:
Thus it is possible that Jones was murdered if and only if it is not necessary that Jones was not murdered.

# Alethic modalities
Necessity and possibility are 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.
A proposition is said to be
- possible if it is not necessarily false (regardless of whether it actually is true or false);
- necessary if it is not possibly false;
- contingent if it is not necessarily false but not necessarily true either. In formal contexts, therefore, contingency refers to a limited case of possibility.

## Logical necessity
There are a number of different alethic modalities: logical possibility is, perhaps, the weakest, since almost anything intelligible is logically possible: Possibly, pigs can fly, Elvis is still alive, and the atomic theory of matter is false.
Likewise, almost nothing is logically impossible: something logically impossible is called a contradiction or a logical falsehood. It is possible that Elvis is alive; but it is impossible that Elvis is alive and is not alive, regardless of what Schrodinger might say. Many logicians also hold that mathematical truths are logically necessary: it is impossible that 2+2 = 4.
Something which is logically necessary is called a logical truth.  For example, it is necessary that if Elvis is alive, then he is alive.

## Physical possibility
Something is physically possible if it is permitted by the laws of nature.  For example, it is possible for there to be an atom with an atomic number of 150, though there may not in fact be one.  On the other hand, it is not possible, in this sense, for there to be an element whose nucleus contains cheese.  While it is logically possible to accelerate beyond the speed of light, it is not, according to modern science, physically possible for objects with mass.

## Metaphysical possibility
Philosophers ponder the properties 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, or that God exists (or does not exist).  Saul Kripke has argued that every person necessarily has the parents they do have: anyone with different parents wouldn't be the same person.
Metaphysical possibility is generally thought to be stronger 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 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.

## Confusion with epistemic modalities
Alethic modalities and epistemic modalities (see below) are often expressed in English using the same words.  Thus, "It is possible that bigfoot exists" might mean either "It would be possible for such a creature as a bigfoot to exist," or (more likely), "As far as I know, there may be some bigfoots."
In the former case, the speaker might know that there are not any bigfoots, but is saying that (unlike round squares), there could be some  the existence of bigfoot is not impossible.  In the latter case he is saying that there may well be some "right now".

# Epistemic logic
Epistemic modalities (from the Greek episteme, knowledge), deal with the certainty of sentences.  The operators are translated as  "It is certainly true that..." and "It may (given the available information) be true that..."  In ordinary speech both modalities are often expressed in similar words; the following contrasts may help:
A person, Jones, might reasonably say both: (1) "No, it is not possible that Bigfoot exists; I am quite certain of that;" and, (2) "Sure, 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 means that things might have been otherwise.  He does not mean "it is possible that Bigfoot exists  for all I know." (So he is not contradicting (1).) Rather, he is making the metaphysical claim that it's possible for Bigfoot to exist, even though he doesn't.
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 that would show that it is not logically possible for Goldbach's conjecture to be falsethere 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 for a mathematical truth to have been false, but (3) only makes a claim about whether it is possible that the mathematical claim turns out false, for all Jones knows, and so again Jones does not contradict himself.  It is worthwhile to observe that Jones is not necessarily correct:  It is possible (epistemically) that Goldbach's conjecture is both true and unprovable.
Epistemic possibilities also bear on the actual world in a way that metaphysical possibilities do not. Metaphysical possibilities bear on ways the world might have been, but epistemic possibilities bear on the way the world may be (for all we know). Suppose, for example, that I want to know whether or not to take an umbrella before I leave. If you tell me "It is possible that it is raining outside"  in the sense of epistemic possibility  then that would weigh on whether or not I take the umbrella. But if you just tell me that "It is possible for it to rain outside"  in the sense of metaphysical possibility  then I am no better off for this bit of modal enlightenment.

# Temporal logic
There are several analogous modes of speech, which though less likely to be confused with alethic modalities are still closely related.  One is talk of time.  It seems reasonable to say that possibly it will rain tomorrow, and possibly it won't; on the other hand, if it rained yesterday, if it really already did so, then it cannot be quite correct to say "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.
A standard method for formalizing talk of time is to use two pairs of operators, one for the past and one for the future. For the past, let "It has always been the case that . . ." be equivalent to the box, and let "It was once the case that . . ." be equivalent to the diamond. For the future, let "It will always be the case that . . ." be equivalent to the box, and let "it will eventually be the case that . . ." be equivalent to the diamond.  If these two systems are used together, it will, obviously, be necessary to indicate, as by subscripts, which box is which.
Additional binary operators are also relevant to temporal logics, q.v. Linear Temporal Logic.

# Deontic logic
Likewise talk of morality, or of obligation and norms generally, seems to have a modal structure.  The difference between "You must do this" and "You may do this" looks a lot like the difference between "This is necessary" and "This is possible."  Such logics are called deontic, from the Greek for "duty".

# Other modal logics
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" or non-truth-functional sentential operators) that make them all varieties of the same thing. Epistemic logic is arguably best captured in the system "S4"; deontic logic in the system "D", temporal logic in "T" and alethic logic arguably with "S5".

# Interpretations of modal logic
In the most common interpretation of modal logic, one considers "all 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.
Whether this "possible worlds idiom" is the best way to interpret modal logic, and how literally this idiom can be taken, is a live issue for metaphysicians. For example, the possible worlds idiom would translate the claim about Bigfoot as "There is some possible world in which Bigfoot exists". To maintain that Bigfoot's existence is possible, but not actual, one could say, "There is some possible world in which Bigfoot exists; but in the actual world, Bigfoot does not exist". But it is unclear what it is that making modal claims commits us to. Are we really alleging the existence of possible worlds, every bit as real as our actual world, just not actual? David Lewis 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 world (see Indexicality). That position is a major tenet of "modal realism". Most philosophers decline to endorse such a view, considering it ontologically extravagant, and preferring to seek various ways to paraphrase away the ontological commitments implied by our modal claims.

# Formal rules
Many systems of modal logic, with widely varying 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.
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 Lewis, much employed since, denotes "necessarily p" by a prefixed "box" ( \Box p ) whose scope is established by parentheses. Likewise, a prefixed "diamond" (\Diamond p) denotes "possibly p." Regardless of notation, each of these operators is definable in terms of the other:
- \Box p (necessarily p) is equivalent to \neg \Diamond \neg p  ("not possible that not-p")
- \Diamond p  (possibly p) is equivalent to  \neg \Box \neg p  ("not necessarily not-p")
Hence \Box and \Diamond form a dual pair of operators.
In many modal logics, the necessity and possibility operators satisfy the following analogs of de Morgan's laws from Boolean algebra:
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. Many modal logics, known collectively as normal modal logics, include the following rule and axiom:
- N, Necessitation Rule: If p is a theorem (of any system invoking N), then \Box p is likewise a theorem.
- K, Distribution Axiom:   \Box (p \rightarrow q)  \rightarrow (\Box p \rightarrow \Box q).
The weakest normal modal logic, named K in honor of Saul Kripke, is simply the propositional calculus augmented by  \Box , 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   \Box p  is true then   \Box \Box 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:   \Box p \rightarrow p  (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 S1^0.
Other well-known elementary axioms are:
- 4:  \Box p \rightarrow \Box \Box p
- B:  p \rightarrow \Box \Diamond p
- D:  \Box p \rightarrow \Diamond p
- E:  \Diamond p \rightarrow \Box \Diamond p.
These axioms yield the systems:
- K := K + N
- T := K + T
- S4 := T + 4
- S5 :=  S4 + B or T + E
- D := K + D.
K through S5 form a nested hierarchy of systems, making up the core of normal modal logic. D is primarily of interest to those exploring the deontic interpretation of modal logic.
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. Although controversial, this is commonly justified on the grounds that S5 is the system obtained if every possible world is possible relative to every other world. Other systems of modal logic have been formulated, in part because S5 does not describe every kind of metaphysical modality of interest. This suggests that talk of possible worlds and their semantics may not do justice to all modalities.

# Development of modal logic
Although Aristotle's logic is almost entirely concerned with the theory of the categorical syllogism, there are passages in his work, such as the famous Sea-Battle Argument in De Interpretatione  9, that are now seen as anticipations of modal logic and its connection with potentiality and time. 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.
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 C. H. Langford), which introduced the five systems S1 through S5. The contemporary era in modal logic 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.
A. N. Prior created temporal logic, closely related to modal logic, in 1957 by adding modal operators  and  meaning "henceforth" and "hitherto." Vaughan Pratt introduced dynamic logic in 1976. In 1977, Amir Pnueli proposed using temporal logic to formalise the behaviour of continually operating concurrent programs. Flavors of temporal logic include propositional dynamic logic (PDL), propositional linear temporal logic (PLTL), linear temporal logic (LTL), computational tree logic (CTL), Hennessy-Milner logic, and T.
The mathematical structure of modal logic, namely Boolean algebras augmented with unary operations (often called "modal algebras"), began to emerge with J. C. C. McKinsey's 1941 proof that S2 and S4 are decidable, and reached full flower in the work of Alfred Tarski 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 and closure operators of topology. Texts on modal logic typically do little more than mention its connections with the study of Boolean algebras and topology. For a thorough survey of the history of formal modal logic and of the associated mathematics, see Goldblatt (2006).