Modal logic

In philosophical logic, a modal logic is any logic for handling modalities: concepts like possibility, impossibility, 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 $$\Box$$ (or L) for Necessarily and $$\Diamond$$ (or M) for Possibly. Each can be defined from the other and negation. For example:


 * $$\Diamond p = \lnot\, \Box\, \lnot\, p.$$

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 alethic modalities, from the Greek aletheia, truth. 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 sentence is said to be Thus if something is necessarily true, then it is true; if it is true, then it is possible.
 * possible if it might be true (regardless of whether it is or is not actually true);
 * necessary if it could not possibly be false;
 * contingent if it is actually true, but not necessarily true. It could have been otherwise, so it is possibly true, and possibly false.

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. 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 travel faster than the speed of light, it is not, according to modern science, physically possible.

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 (fewer things are 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 false&mdash;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 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.

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" (sic:lowercase) 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, then asserting that possible worlds are as real as our own. This position is called "modal realism". Unsurprisingly, most philosophers decline to sign on to this ontologically extravagant doctrine, preferring to seek various ways to paraphrase away the ontological commitments implied by our modal claims.

Formal rules
Many modal logics, with widely varying properties, have been proposed since C. I. Lewis began working in the area in 1912. Hughes and Cresswell (1996), for example, describe 42 normal and 25 non-normal modal logics. Zeman (1973) describes some systems Hughes and Cresswell omit.

Modern treatments of modal logic begin by augmenting the propositional calculus 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 each other: Hence $$\Box$$ and $$\Diamond$$ form a dual pair of operators.
 * $$\Box p$$ (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")

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 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: If $$ \Box (p \rightarrow q)$$ then $$ \Box p \rightarrow \Box q.$$

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 resolve 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. This may not be a great defect for K, since such questions seem rather forced, and any attempt to answer them involves us in confusing issues. 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 in fact the case. The axiom T remedies this defect: Other well-known elementary axioms are: These axioms yield the systems: 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.
 * 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.
 * 4: $$ \Box p \rightarrow \Box \Box p$$
 * B: $$ p \rightarrow \Box \Diamond p$$
 * D: $$ \Box p \rightarrow \Diamond p$$
 * E: $$ \Diamond p \rightarrow \Box \Diamond p.$$
 * K := K + N
 * T := K + T
 * S4 := T + 4
 * S5 := S4 + B or T + E
 * D := K + D.

The commonly employed system S5 robustly addresses the usual modal perplexities by making all modal truths necessary. For example, if it is possible that p, then it is necessarily possible that p. Also, if it is necessary that p, it is also necessary that it's necessary. This is commonly justified on the grounds that S5 is the system obtained if every possible world is possible relative to every other world. Nevertheless, 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 [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 Boolean algebra and topology. For a thorough survey of the history of formal modal logic and of the associated mathematics, see Goldblatt (2006).

Intensionality and modal logic
Some people argue that modal logics are characterized by semantic intensionality: the truth value of a complex formula cannot be determined by the truth values of its subformulae, and modal operators cannot be formalized by an extensional semantics: both "George W. Bush is President of the United States" and "2 + 2 = 4" are true, yet "Necessarily, George W. Bush is President of the United States" is false, while "Necessarily, 2 + 2 = 4" is true.

Actually, this claim is not correct, since we can give the semantics of a modal logic by structural induction, if we use stateful models, also called coalgebraic models. For example, we can consider the following very simple modal logic syntax:

$$F ::= \Diamond F | F \land F | \lnot F | \mathrm{true}$$

We can derive dual connectives using the basic ones:

$$\mathrm{false} = \lnot \mathrm{true}$$

$$ \Box F = \lnot (\Diamond \lnot F)$$

$$ F_1 \lor F_2 = \lnot (\lnot F_1 \land \lnot F_2)$$

The truth value of a formula is defined over models that are not sets, but transition systems.

A transition system is a pair $$(S,T)$$ where $$S$$ is a set and $$T \subseteq S \times S$$.

The interpretation of the logic over the state $$s \in S$$, given a transition system $$(S,T)$$, is a relation $$\models \subseteq S \times F$$, where $$s \models F$$ is read "the state s satisfies the formula F", given by structural induction as follows:

$$s \models \lnot F \iff \mathrm{not}\,s \models F$$

$$s \models F_1 \land F_2 \iff s \models F_1\, \mathrm{and}\, s \models F_2$$

$$s \models \Diamond F \iff \exists s_1. (s,s_1) \in T \, \mathrm{and}\, s_1 \models F$$

If we view a transition system $$(S,T)$$ as a set $$S$$ of states and a set $$T$$ of transitions from a state to another, the modal formula $$\Diamond F$$, which is called the "next" modality, is read as "in my possible next states, there is one that satisfies F".

This logic is too simple for practical uses; more complicated logics can have more complicated models (an example being Kripke frames), however the definition of the semantics is usually given by structural induction over states.

Acknowledgements
This article includes material from the Free On-line Dictionary of Computing, used with permission under the GFDL.

منطق طوري Modallogik Modallogik Lógica modal Logique modale Modala logiko לוגיקה מודלית Modale logica 様相論理学 Modallogikk Modallogikk Модальная логика Modal logik 模态逻辑