Modal 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 modal sentential 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. They are defined in terms of one another this way:

\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

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.

Thus if something is necessarily true, then it is true; if it is true, then it is possible.

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—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 infamously bit the bullet and said yes, possible worlds are as real as our own. This position is called "modal realism". Unsurprisingly, most philosophers are unwilling to sign on to this particular doctrine, seeking alternate ways to paraphrase away the apparent ontological commitments implied by our modal claims.

Formal rules

There are many modal logics, with many different properties; see Cresswell and Hughes (1996) for a survey.

Modal logics add to the well formed formulae of propositional logic operators for necessity and possibility. The notation of Clarence Irving Lewis, often employed to this day, represents "necessarily p" using a prefixed "box" ( $\Box p$ ). Likewise, a prefixed "diamond" ( $\Diamond p$ ) represents "possibly p." Whatever the notation, each of these operators is definable in terms of each 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)

The $\Box$ and $\Diamond$ are said to form a dual pair of operators.

In many such 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 propositional logic to create a usable system of modal logic has been much debated. One weak system, named K in honor of Saul Kripke, adds merely the following to any axiomatization of propositional logic:

Necessitation Rule: If p is a theorem of K, then so is $\Box p$ .
Distribution Axiom: If $\Box (p\rightarrow q)$ then $(\Box p\rightarrow \Box q)$ (this is also known as axiom K)

These rules lack an axiom to go from the necessity of p to p actually being the case, and therefore are usually supplemented with the following "reflexivity" axiom, which yields a system often called T.

Reflexivity Axiom: $\Box p\rightarrow p$ (If p is necessary, then p is the case)

This is a rule of most, but not all modal logics. Zeman (1973) covers systems such as S1^0 that lack this rule.

K is a weak modal logic. In particular, whether a proposition can be necessary but only contingently necessary is unresolved. 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, varying answers to such questions yield varying systems of modal logic.

A commonly employed system, S5, robustly answers such questions by adding axioms which make 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 of interest either.

Development of modal logic

Although Aristotle's logic is almost entirely concerned with the theory of the categorical syllogism, his work also contains some extended arguments on points of modal logic (such as his famous Sea-Battle Argument in De Interpretatione § 9) and their connection with potentialities and time. The Scholastics, in particular William of Ockham and John Duns Scotus, laid the ground for a rigorous theory of modal logic, mostly within the context of the logic of 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 mathematical structure of modal logic, namely Boolean algebras augmented with operators, emerged in the work of J. C. C. McKinsey, Alfred Tarski, and Tarski's students, beginning with McKinsey's 1941proof that S2 and S4 are decidable. Interior algebra discusses the algebraic structure of S4 and S5. In 1959, Saul Kripke introduced the now-standard relational or possible worlds semantics for modal logics. Vaughan Pratt introduced dynamic logic in 1976.

A. N. Prior created temporal logic, closely related to modal logic, in 1957 by adding modal operators [F] and [P] meaning "henceforth" and "hitherto." In 1977, Amir Pnueli proposed using temporal logic to formalise the behaviour of continually operating concurrent programs. Other flavours of modal logic include: propositional dynamic logic (PDL), propositional linear temporal logic (PLTL), linear temporal logic (LTL), computational tree logic (CTL), Hennessy-Milner logic, and T.

A note about intensionality of modal logics

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:

$F::=\Diamond F|F\land F|\lnot F|true$

We can derive dual connectives using the basic ones:

$false=\lnot true$

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

$F_{1}\lor F_{2}=\lnot (\lnot F_{1}\land \lnot F_{2})$

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:

$s\models \lnot F\iff not\,s\models F$

$s\models F_{1}\land F_{2}\iff s\models F_{1}\,and\,s\models F_{2}$

$s\models \Diamond F\iff \exists s_{1}.(s,s_{1})\in T\,and\,s_{1}\models F$

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.

References

Patrick Blackburn, Maarten de Rijke, and Yde Venema (2001) "Modal Logic". Cambridge University Press.
Aleksander Chagrov and Michael Zakharyaschev (1997) "Modal Logic". Oxford University Press.
Brian F. Chellas (1980) "Modal Logic: an introduction". Cambridge University Press.
M. Fitting and R.L. Mendelsohn (1998) First Order Modal Logic. Kluwer Academic Publishers.
Rod Girle (2000) Modal Logics and Philosophy. Acumen (UK). The proof theory employs refutation trees (semantic tableaux). A good introduction to the varied interpretations of modal logic.
Robert Goldblatt (1992) "Logics of Time and Computation", 2nd ed. CSLI Lecture Notes No. 7. CSLI, distributed by the University of Chicago Press.
------ (1993) "Mathematics of Modality", CSLI Lecture Notes No. 43. CSLI; distributed by the University of Chicago Press.
G.E. Hughes and M.J. Cresswell (1968) An Introduction to Modal Logic. Methuen.
------ (1984): A Companion to Modal Logic. Medhuen.
------ (1996): A New Introduction to Modal Logic. Routledge.
E.J. Lemmon (with Dana Scott), 1977: An Introduction to Modal Logic, American Philosophical Quarterly Monograph Series, no. 11 (ed. by Krister Segerberg). Basil Blackwell.
Zeman, J. J. (1973) Modal Logic. Reidel.

External links

Stanford Encyclopedia of Philosophy: Modal logic -- James Garson.
A discussion of modal logic by John McCarthy
Bibliography of Non-Standard Logics by Peter Suber
List of Logic Systems List of most of the more popular modal logics.
Advances in Modal Logic (bi-annual international conference and book series in Modal Logic)
Basic Concepts in Modal Logic (pdf) by Edward N. Zalta

Acknowledgements

This article contains some material originally from the Free On-line Dictionary of Computing which is used with permission under the GFDL.