 Modal logic

Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing 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 modalities, or modalities of truth, include possibility ("Possibly, p", "It is possible that p"), necessity ("Necessarily, p", "It is necessary that p"), and impossibility ("It is impossible that p")^{[1]}. Other modalities that have been formalized in modal logic include temporal modalities, or modalities or 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")^{[2]} ^{[3]}, deontic modalities (notably, "It is obligatory that p", and "It is permissible that p"), epistemic modalities, or modalities of knowledge ("It is known that p")^{[4]} and doxastic modalities, or modalities of belief ("It is believed that p").^{[5]}
A formal modal logic represents modalities using modal operators. 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.
The basic unary (1place) modal operators are usually written for Necessarily and for Possibly. In a classical modal logic, each can be expressed by the other with negation:
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.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 seabattle 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 selfaware 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.
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".
The contemporary era in modal semantics began in 1959, when Saul Kripke (then only a 19yearold Harvard University undergraduate) introduced the nowstandard 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 semantictableaus or analytic tableaux, as explained by 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 "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 Robert Goldblatt (2006).
Formalizations
Semantics
The semantics for modal logic are usually given like so:^{[6]} First we define a frame, which consists of a nonempty 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, <G, R>.
Next, the frame is extended to a model by specifying the truthvalues of all propositions at each of the worlds in G. We do so by defining a relation ⊨ between possible worlds and propositional letters. If there is a world w such that w ⊨ P, then P is true at w. A model is thus an ordered triple, <G, R, ⊨>.
Then we recursively define the truth of a formula in a model:
 w ⊨ ¬P if and only if w P
 w ⊨ (P Q) if and only if w ⊨ P and w ⊨ Q
 w ⊨ P if and only if for every element v of G, if w R v then v ⊨ P
 w ⊨ P if and only if for some element v of G, it holds that w R v and v ⊨ P
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.
It should also be noted that the definition of makes vacuously true certain sentences, since when it speaks of "every world that is accesible 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 is true.
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:
 reflexive iff w R w, for every w in G
 symmetric iff w R v implies v R w, for all w and v in G
 transitive iff w R v and v R q together imply w R q, for all w, v, q in G.
 serial iff, for each w in G there is some v in G such that w R v.
 euclidean iff, for every u,v and w, u R w and v R w implies u R v (note that it also implies: v R u)
The logics that stem from these frame conditions are:
 K := no conditions
 D := serial
 T := reflexive
 S4 := reflexive and transitive
 S5 := reflexive, symmetric, transitive and Euclidean
S5 models are reflexive transitive and euclidean. 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. 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 edge (relation) can be added).
For example, in S4:

 w ⊨ P if and only if for some element v of G, it holds that v ⊨ P and w R v.
However, in S5, we can just say that

 w ⊨ P if and only if for some element v of G, it holds that v ⊨ P.
We can drop the accessibility clause from the latter stipulation because it is trivially true of all S5 frames that w R v.
All of these logical systems can also be defined axiomatically, as is shown in the next section. For example, in S5, the axioms P → P, P → P, and 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.
Axiomatic systems
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 nonnormal 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 C. I. Lewis, much employed since, denotes "necessarily p" by a prefixed "box" ( ) whose scope is established by parentheses. Likewise, a prefixed "diamond" () denotes "possibly p". Regardless of notation, each of these operators is definable in terms of the other:
 (necessarily p) is equivalent to ("not possible that notp")
 (possibly p) is equivalent to ("not necessarily notp")
Hence and 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:
 "It is not necessary that X" is logically equivalent 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 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 logics, include the following rule and axiom:
 N, Necessitation Rule: If p is a theorem (of any system invoking N), then is likewise a theorem.
 K, Distribution Axiom: .
The weakest normal modal logic, named K in honor of Saul Kripke, is simply the propositional calculus 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 is true then 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 wellknown 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 S1^{0}.
Other wellknown elementary axioms are:
 4:
 B:
 D:
 5:
These yield the systems (axioms in bold): K := K + N
 T := K + T
 S4 := T + 4
 S5 := S4 + 5
 D := K + D.
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, (If it ought to be that p, then it is permitted that p) seems appropriate, but we should probably not include that . 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).
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.
Alethic logic
Main article: Alethic modalityModalities 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.
In classical modal logic, a proposition is said to be
 possible if and only if it is not necessarily false (regardless of whether it is actually true or actually false);
 necessary if and only if it is not possibly false; and
 contingent if and only if it is not necessarily false and not necessarily true (i.e. possible but not necessarily true).
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.
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) 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.
Physical possibility
Something is physically possible if it is permitted by the laws of physics. For example, current theory allows for there to be an atom with an atomic number of 150, though there may not in fact be any such atoms in existence. Similarly, while it is logically possible to accelerate beyond the speed of light, modern science stipulates that it is not physically possible for material particles or information.
Metaphysical possibility
Philosophers 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. Saul Kripke has argued that every person necessarily has the parents they do have: anyone with different parents would not be the same person.^{[7]}
Metaphysical possibility is generally thought 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 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. "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).
Epistemic logic
Main article: Epistemic logicEpistemic modalities (from the Greek episteme, knowledge), deal with the certainty of sentences. The operator is translated as "x knows that…", and the operator is translated as "For all x knows, it may be true that…" In ordinary speech both metaphysical and epistemic modalities are often expressed in similar words; the following contrasts may help:
A person, Jones, might reasonably say both: (1) "No, it is not possible that Bigfoot exists; I am quite certain of that"; and, (2) "Sure, 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)).
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.^{[8]}
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 be an axiom in these systems? While the answer to this question is unclear, there is at least one axiom that must be included in epistemic modal logic, because it is minimally true of all modal logics (see the section on axiomatic systems):
 K, Distribution Axiom: .
But this is disconcerting, because with K, we can prove that we know all the logical consequences of our beliefs: If q is a logical consequence of p, then . And if so, then we can deduce that 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 Kripkean modal interpretations of epistemic cases where 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 8683317618811886495518194401279999999 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 logic is an idealized account of knowledge, and explains objective, rather than subjective knowledge (if anything).
Temporal logic
Main article: Temporal logicTemporal 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:
 FP : It will sometimes be the case that P
 GP : It will always be the case that P
 PP : It was sometime the case that P
 HP : 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,
 = P is the case at some time t
 = P is the case at every time t
Or we can trade these operators to deal only with the future (or past). For example,
 = FP
 = GP
or,
 = P and/or FP
 = P and GP
The operators F and G may seem initially foreign, but they create normal modal systems. Note that FP is the same as ¬G¬P. We can combine the above operators to form complex statements. For example, PP → PP 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, seeing as how 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, 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.
Additional binary operators are also relevant to temporal logics, q.v. Linear Temporal Logic.
Versions of temporal logic can be used in computer science to model computer operations and prove theorems about them. In one version, means "at a future time in the computation it is possible that the computer state will be such that P is true"; means "at all future times in the computation P will be true". In another version, means "at the immediate next state of the computation, P might be true"; 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 notfullyunderstood computations; there are many other modal logics specialized to different types of program analysis. Each one naturally leads to slightly different axioms.
A variation, closely related to Temporal or Chronological or Tense logics, are Modal logics based upon "topology", "place", or "spatial position". ^{[9]}^{[10]} One might also take note that in the Russian language, verbs have an aspect, based commonly upon time, but position also.
Deontic logic
Main article: Deontic logicLikewise 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".
Deontic logics commonly lack the axiom T semantically corresponding to the reflexivity of the accessibility relation in Kripke semantics: in symbols, . 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, 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.^{[11]}
One other principle that is often (at least traditionally) accepted as a deontic principle is D, , 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 and formalize ethics with standard modal logic, we run into some problems. Suppose that we have a proposition K: you have killed the victim, and another, Q: you have killed the victim quickly. Now suppose we want to express the thought that "if you have killed the victim, it ought to be the case that you have killed him quickly". There are two likely candidates,
 (1)
 (2)
But (1) says that if you have killed the victim, then it ought to be the case that you have killed him quickly. This surely isn't right, because you ought not to have killed him at all. And (2) doesn't work either. If the right representation of "if you have killed the victim then you ought to have killed him quickly" is (2), then the right representation of (3) "if you have killed the victim then you ought to have killed him slowly" is . Now suppose (as seems reasonable) that you should not have killed the victim, or . But then we can deduce , which would express sentence (3). So if you should not have killed the victim, then if you did kill him, you should have killed him slowly. But that can't be right, and is not right when we use natural language. Telling someone they should not kill the victim certainly does not imply that they should kill the victim slowly if they do kill him.^{[12]}
Doxastic logic
Main article: Doxastic logicDoxastic 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 , often written "B", to mean "It is believed that", or when relativized to a particular agent s, "It is believed by s that".
Other modal logics
See also: Intensional logicSignificantly, 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.
The ontology of possibility
Further information: Accessibility relation and Possible worldsIn 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.
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.^{[13]} 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.^{[14]} David Lewis, on the other hand, made himself notorious by biting the bullet, asserting that all merely possible worlds are as real as our own, and that what distinguishes our world as actual is simply that it is indeed our world – this world.^{[15]} That position is a major tenet of "modal realism". Some philosophers decline to endorse any version of modal realism, considering it ontologically extravagant, and prefer to seek various ways to paraphrase away these ontological commitments. Robert Adams holds that 'possible worlds' are better thought of as 'worldstories', or consistent sets of propositions. Thus, it is possible that you rolled a 4 if such a state of affairs can be described coherently.^{[16]}
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".
Applications
 Modality has also been treated from the viewpoint of "counterfactuals" in literature (see Victorian Studies). ^{[17]}
^{[18]} ^{[19]}
 Modality modifies propositions and modalities provide closure (i.e.: propositions with modalities are still propositions). Thus, as propositions constitute a part of language, they may be understood as subject to linguistic analysis such as that of Noam Chomsky. Modalities might then be viewed as being contextfree, contextsensitive, or even fully phrasestructured (Chomsky type0) languages. This broadens the view of modalities which are usually viewed as contextfree. A discussion of this may be found under the Philosophy of language.
As an example, consider the following contextsensitive grammar for an alethic Modal logic grammar:
L=({S, X, R, M}, {Φ, , , (, ), &, , &, v, ~}, P, S), where Φ is a metavariable for propositions, and P are the following production rules:
S XR  M
XR M  M
M T M  T
T (T T)  (T & T)  (T v T)  (~T)  Φ
A sample derivation:
S XR M T ( T & T ) ( Φ & T ) ( p & T ) ( p & Φ ) ( p & q )
Of course, an equivalent contextfree language is easy to construct, but the point here is that a Modal logic based upon a contextsensitive grammar is possible. More complex Modal logics which are Chomsky type0 or type1 extends the philosophical discussion of Modal logics.
 Aristotle classified and discussed rhetoric as being based upon the enthymeme, thus closely related to logic. However, it is clear that if logic is extended by modal logics, multivalued logics, etc., then rhetoric must also be extended by modern developments.
Thus, for example, the following might be persuasive rhetorically:
(p & q)
______________________________
Therefore p
but ~[ (p & q) ( p & q) ]
Further applications
Modal logics have begun to be used in area of the humanities such as literature, poetry, art and history.^{[20]}^{[21]}^{[22]}
Controversies
Modal logic has been rejected by many philosophers. (Historically, philosophers starting with Aristotle seem to have had priority of interest in modal logic over mathematicians.) The primary reason for this rejection is that modality provides a different logical framework with which to analyze society. Nicholas Rescher has pointed out how radically conservative were the views of Bertrand Russell in his unreasoned rejection of both Modal Logic and the ideas of the philosopher Alexius Meinong. Indeed, Arthur Norman Prior warned his protégé Ruth Barcan to prepare well in the debates concerning Modal Logic (with Willard Van Orman Quine) due to the biases against Modal Logic.^{[23]}
See also
Notes
 ^ "Formal Logic", by A. N. Prior, Oxford Univ. Press, 1962, p. 185
 ^ "Temporal Logic", by Rescher and Urquhart, SpringerVerlag, 1971, p. 52
 ^ "Past, Present and Future", by A. N. Prior, Oxford Univ. Press, 1967
 ^ "Knowledge and Belief", by Jaakko Hinntikka, Cornell Univ. Press, 1962
 ^ "Topics in Philosophical Logic", by N. Rescher, Humanities Press, 1968, p. 41
 ^ Fitting and Mendelsohn. FirstOrder Modal Logic. Kluwer Academic Publishers, 1998. Section 1.6
 ^ Saul Kripke. Naming and Necessity. Harvard University Press, 1980. pg 113
 ^ See Goldbach's conjecture > Origins
 ^ Nicholas Rescher, James Garson, "Topological Logic" in The Journal of Symbolic Logic, 33(4):537548, December, 1968
 ^ Georg Henrik von Wright, "A Modal Logic of Place", in E. Sosa (Editor), pp. 6573, "The Philosophy of Nicholas Rescher: Discussion and Replies", D. Reidel, Dordrecht, Holland, 1979
 ^ See, e.g., Sven Hansson, "Ideal Worlds—Wishful Thinking in Deontic Logic", Studia Logica, Vol. 82 (3), pp. 329–336, 2006.
 ^ Ted Sider's Logic for Philosophy, unknown page.
 ^ Kripke, Saul. Naming and Necessity. (1980; Harvard UP), pp. 43–5.
 ^ Kripke, Saul. Naming and Necessity. (1980; Harvard UP), pp. 15–6.
 ^ David Lewis, On the Plurality of Worlds (1986; Blackwell)
 ^ Adams, Robert M. Theories of Actuality. Noûs, Vol. 8, No. 3 (Sep., 1974), particularly pp. 225–31.
 ^ "Possible Worlds of Fiction and History", by Dolezel, Lubomír, New Literary History, 1998, 29(4): 785809
 ^ "Lives Unled in Realist Fiction", by Miller, Andrew H., Representations 98, Spring 2007, The Regents of the University of California, ISSN 1553855X, pp. 118134.
 ^ "Not Forthcoming", by Miller, Andrew H., Dickens Universe, 2009, Univ. of California, Santa Cruz
 ^ See http://www.estherlederberg.com/EImages/Extracurricular/Dickens%20Universe/Counter%20Factuals.html
 ^ Andrew H. Miller, "Lives Unled in Realist Fiction", Representations 98, Spring 2007, The Regents of the University of California, ISSN 1553855X, pp. 118134
 ^ See also http://www.estherlederberg.com/EImages/Extracurricular/Dickens%20Universe/Page%2017%20CounterFactuals.html
 ^ "Modalities: Philosophical Essays", by Ruth Barcan Marcus, Oxford Univ. Press, 1993, Chapter 14
References
 This article includes material from the Free Online Dictionary of Computing, used with permission under the GFDL.
 Blackburn, Patrick; de Rijke, Maarten; and Venema, Yde (2001) Modal Logic. Cambridge University Press. ISBN 0521802008
 BarcanMarcus, 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 30942. Reprinrted in Jaakko Intikka (ed.) The Philosophy of Mathematics, Oxford University Press, 1969 (Semantic Tableaux proof methods)
 Beth, Evert W., "Formal Methods: An Introduction to Symbolic Logic and to the Study of Effective Operations in Arithmetic and Logic", D. Reidel, 1962 (Semantic Tableaux proof methods)
 Blackburn, P.; van Benthem, J.; and Wolter, Frank; Eds. (2006) Handbook of Modal Logic. North Holland.
 Chagrov, Aleksandr; and Zakharyaschev, Michael (1997) Modal Logic. Oxford University Press. ISBN 0198537794
 Chellas, B. F. (1980) Modal Logic: An Introduction. Cambridge University Press. ISBN 0521224764
 Cresswell, M. J. (2001) "Modal Logic" in Goble, Lou; Ed., The Blackwell Guide to Philosophical Logic. Basil Blackwell: 136–58. ISBN 0631206930
 Fitting, Melvin; and Mendelsohn, R. L. (1998) First Order Modal Logic. Kluwer. ISBN 0792353358
 James Garson (2006) Modal Logic for Philosophers. Cambridge University Press. ISBN 0521682290. 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). ISBN 0773521399. Proof by refutation trees. A good introduction to the varied interpretations of modal logic.
 Goldblatt, Robert (1992) "Logics of Time and Computation", 2nd ed., CSLI Lecture Notes No. 7. University of Chicago Press.
 —— (1993) Mathematics of Modality, CSLI Lecture Notes No. 43. University of Chicago Press.
 —— (2006) "Mathematical Modal Logic: a View of its Evolution", in Gabbay, D. M.; and Woods, John; Eds., Handbook of the History of Logic, Vol. 6. Elsevier BV.
 Goré, Rajeev (1999) "Tableau Methods for Modal and Temporal Logics" in D'Agostino, M.; Gabbay, Dov; 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. ISBN 0415125995
 Jónsson, B. and Tarski, A., 1951–52, "Boolean Algebra with Operators I and II", American Journal of Mathematics 73: 891939 and 74: 129–62.
 Kracht, Marcus (1999) Tools and Techniques in Modal Logic, Studies in Logic and the Foundations of Mathematics No. 142. North Holland.
 Lemmon, E. J. (with Scott, D.) (1977) An Introduction to Modal Logic, American Philosophical Quarterly Monograph Series, no. 11 (Krister Segerberg, series ed.). Basil Blackwell.
 Lewis, C. I. (with Langford, C. H.) (1932). Symbolic Logic. Dover reprint, 1959.
 Prior, A. N. (1957) Time and Modality. Oxford University Press.
 Snyder, D. Paul "Modal Logic and its applications", Van Nostrand Reinhold Company, 1971 (proof tree methods)
 Zeman, J. J. (1973) Modal Logic. Reidel. Employs Polish notation.
Further reading
 D.M. Gabbay, A. Kurucz, F. Wolter and M. Zakharyaschev, ManyDimensional Modal Logics: Theory and Applications, Elsevier, Studies in Logic and the Foundations of Mathematics, volume 148, 2003, ISBN 0444508260. 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.
External links
 Stanford Encyclopedia of Philosophy:
 "Modal logic" – by James Garson.
 "Provability Logic" – by Rineke Verbrugge.
 Edward N. Zalta, 1995, "Basic Concepts in Modal Logic."
 John McCarthy, 1996, "Modal Logic."
 Molle a Java prover for experimenting with modal logics
 Suber, Peter, 2002, "Bibliography of Modal Logic."
 Marcus, Ruth Barcan, "Modalities" OUP 1993, 1995
 List of Logic Systems 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
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