Modal logic, or (less commonly) intensional logic is the branch of logic that deals with sentences that are qualified by modalities such as can, could, might, may, must, possibly, and necessarily, and others. Any logical system making use of modal operators, such as possibly, or necessarily is thus also called a modal logic. Modal logics are characterized by semantic intensionality: non-modal logics all have the feature that the truth value of a complex sentence is determined by the truth values of its sub-sentences. They are thus extensional. In modal logics, by contrast, this does not hold: 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. Logic (from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy amongst philosophers (see below). ... George Walker Bush (born July 6, 1946) is the current President of the United States and former Governor of the State of Texas. ...

A formal modal logic represents modalities using modal sentential operators. The basic set of modal operators are usually given to be and . In alethic modal logic (i.e. the logic of necessity and possibility) the represents necessity and the possibility. A sentence is said to be In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ...

• possible if it might be true (whether it is actually true or actually false);
• necessary if it could not possibly be false;
• contingent if it is not necessarily true, i.e., is possibly true, and possibly false. A contingent truth is one which is actually true, but which could have been otherwise.

Metaphysical and other modalities

Subjunctive, epistemic

Modal logic is most often used for talk of the so-called alethic modalities: "it is necessarily the case that..." or "it is possibly the case that...." These (which include metaphysical modalities and logical modalities) are most easily confused with epistemic modalities (from the Greek episteme, knowledge): "It is certainly true that..." and "It may (given the available information) be true that..." In ordinary speech both modalities are often expressed with the same words; the following contrasts may help: Subjunctive possibility (also called aletheic possibility or metaphysical possibility) is the form of modality most frequently studied in modal logic. ...

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 subjunctive 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. In mathematics, Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ...

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.

The vast bulk of philosophical literature on modalities concerns subjunctive rather than epistemic modalities. (Indeed, most of it concerns the broadest sort of subjunctive modality--that is, bare logical possibility). This is not to say that subjunctive possibilities are more important to our everyday life than epistemic possibilities (consider the example of deciding whether or not to take an umbrella). It is just to say that the priorities in philosophical investigations have not been set by importance to everyday life. Philosophers generally consider logical possibility to be the broadest sort of subjunctive possibility in modal logic. ...

Deontic, temporal

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. Many philosophers and logicians think this reasoning is not very good; but the fact remains that we often talk this way and it is good to have a logic to capture its structure. 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". An obligation can be legal or moral. ... Norms are a sort of sentences or sentence meanings, the most common of which are commands and permissions. ... Deontic logic, first put forward by Ernst Mally in 1926, is a form of modal logic used to describe and reason about obligation and permission. ...

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". In logic, the term temporal logic is used to describe any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. ...

The interpretation 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. Philosophers generally consider logical possibility to be the broadest sort of subjunctive possibility in modal logic. ... In philosophy and logic, the concept of possible worlds is used to express modal claims, claims that involve notions of possibility or necessity. ...

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. David K. Lewis David Kellogg Lewis (September 28, 1941 – October 14, 2001) is considered to have been one of the leading analytic philosophers of the latter half of the 20th century. ... Modal realism is the view, notably propounded by David Lewis, that possible worlds are as real as the actual world. ...

Formal rules

There are many modal logics, with many different properties. In many of them the concepts of necessity and possibility satisfy the following de Morganesque relationship: In logic, De Morgans laws (or De Morgans theorem) are rules in formal logic relating pairs of dual logical operators in a systematic manner expressed in terms of negation. ...

"It is not necessary that X" is equivalent to "It is possible that not X".

"It is not possible that X" is equivalent to "It is necessary that not X".

Although modal logic texts like Hughes and Cresswell's "A New Introduction to Modal Logic" cover some systems where this isn't true.

Modal logic adds to the well formed formulae of propositional logic operators for necessity and possibility. In some notations "necessarily p" is represented using a "box" ( ), and "possibly p" is represented using a "diamond" (). Whatever the notation, the two operators are definable in terms of each other: The propositional calculus is a formal deduction system whose atomic formulas are propositional variables. ...

• (necessarily p) is equivalent to (not possible that not-p)
• (possibly p) is equivalent to (not necessarily not-p)

Hence, the and are called dual operators.

Precisely what axioms must be added to propositional logic to create a usable system of modal logic has been the subject of much debate. One weak system, named K after Saul Kripke, adds only the following to a classical axiomatization of propositional logic: Saul Kripke in 1983 Saul Aaron Kripke (b. ...

• Necessitation Rule: If p is a theorem of K, then so is .
• Distribution Axiom: If then (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. A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...

• (If it's necessary that p, then p is the case)

This is a rules of most, but not all modal logic systems. Jay Zeman's book "Modal Logic" covers systems like S1^0 that don't have this rule.

K is a weak modal logic, however. In particular, it leaves it open that a proposition 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. This may not be a great defect for K, since these seem like awfully strange questions and any attempt to answer them involves us in confusing issues. In any case, different solutions to questions such as these produce different systems of modal logic.

The system most commonly used today is modal logic S5, which robustly answers the questions by adding axioms which make all modal truths necessary: for example, if it's possible that p, then it's necessarily possible that p, and if it's necessary that p it's also necessary that it's necessary. This has been thought by many to be justified on the grounds that it is the system which is obtained when we demand that every possible world is possible relative to every other world. Nevertheless, other systems of modal logic have been formulated, in part, because S5 may not be a good fit for every kind of metaphysical modality of interest to us. (And if so, that may mean that possible worlds talk is not a good fit for these kinds of modality either.)

Development of the field 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 with time. Following on his works, the Scholastics developed the groundwork for a rigorous theory of modal logic, mostly within the context of commentary on the logic of statements about essence and accident. Among the medieval writers, some of the most important works on modal logic can be found in the works of William of Ockham and John Duns Scotus. Aristotle (sculpture) Aristotle (Greek: Αριστοτέλης AristotelÄ“s; 384 BC – March 7, 322 BC) was an ancient Greek philosopher. ... Wikipedia does not yet have an article with this exact name. ... De Interpretatione or Hermeneutics (Peri Hermeneias) is a work of the ancient Greek philosopher Aristotle, mainly on the philosophy of language. ... Scholastic redirects here. ... In philosophy, essence is the attribute (or set of attributes) that make an object or substance what it fundamentally is. ... In philosophy, an accident is a property that its bearer has contingently—that is, a property which its bearer could have failed to have (without having failed to exist), had things been different. ... William of Ockham (also Occam or any of several other spellings) (ca. ... John Duns Scotus (c. ...

The contemporary logical analysis of modality can be traced to C. I. Lewis's "A Survey of Symbolic Logic" (1918), in which he developed the logical systems S1-S5. J. C. C. McKinsey used algebraic methods (Boolean algebras with operators) to prove the decidability of Lewis' S2 and S4 in 1941. Saul Kripke developed the relational semantics for modal logics (1959, 1963). Vaughan Pratt introduced dynamic logic in 1976. Amir Pnueli proposed the use of temporal logic to formalise the behaviour of continually operating concurrent programs in 1977. Clarence Irving Lewis (April 12, 1883 _ February 3, 1964) was a pragmatist philosopher. ... Saul Kripke in 1983 Saul Aaron Kripke (b. ... Kripke semantics (also known as possible world semantics, relational semantics, or frame semantics) is a formal semantics for modal logic systems, created in late 1950s and early 1960s by Saul Kripke. ... Vaughan Pratt is Professor Emeritus of Computer Science at Stanford University. ... In digital electronics, dynamic logic is sometimes used to refer to a class of design assumptions more commonly known as clocked logic, used to distinguish this type of logic from static logic. ... Amir Pnueli (born April 22, 1941) is an Israeli computer scientist who received the Turing Award in 1996 for seminal work introducing temporal logic into computing science and for outstanding contributions to program and systems verification. Born in Nahalal, Israel, Pnueli received a Bachelors degree in Mathematics at the...

Temporal logic is closely related to modal logic, as adding modal operators [F] and [P], meaning, respectively, henceforth and hitherto, leads to a system of temporal logic. In logic, the term temporal logic is used to describe any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. ...

Flavours of modal logics include: propositional dynamic logic (PDL), propositional linear temporal logic (PLTL), linear temporal logic (LTL), computational tree logic (CTL), Hennessy-Milner logic, S1-S5, and T. Linear temporal logic (LTL) is a field of mathematical logic that is able to talk about the future of paths. ... Computational tree logic (CTL) is a temporal logic. ...

References

• M. Fitting and R.L. Mendelsohn, "First Order Modal Logic", Kluwer Academic Publishers, 1998.
• James Garson, 2003. Modal logic. Entry in the Stanford Encyclopedia of Philosophy.
• Robert Goldblatt, "Logics of Time and Computation", CSLI Lecture Notes No. 7, Centre for the Study of Language and Information, Stanford University, Second Edition, 1992, (distributed by University of Chicago Press).
• Robert Goldblatt, "Mathematics of Modality", CSLI Lecture Notes No. 43, Centre for the Study of Language and Information, Stanford University, 1993, (distributed by University of Chicago Press).
• G.E. Hughes and M.J. Cresswell, "An Introduction to Modal Logic", Methuen, 1968.
• G.E. Hughes and M.J. Cresswell, "A Companion to Modal Logic", Medhuen, 1984.
• G.E. Hughes and M.J. Cresswell, "A New Introduction to Modal Logic", Routledge, 1996.
• E.J. Lemmon (with Dana Scott), "An Introduction to Modal Logic", American Philosophical Quarterly Monograph Series, no. 11 (ed. by Krister Segerberg), Basil Blackwell, Oxford, 1977.
• J. Jay Zeeman, 1973. Modal Logic. Clarendon Press (OUP).

The Stanford Encyclopedia of Philosophy (hereafter SEP) is a free online encyclopedia of philosophy run and maintained by Stanford University. ...

See also

• De dicto and de re
• Hybrid logic
• Interior algebra
• Interpretability logic
• Provability logic
• Kripke semantics

De dicto and de re are two phrases used to mark important distinctions in intensional statements, associated with the intensional operators in many such statements. ... Hybrid logic is a form of formal logic which extends modal logic with constructs allowing semantical features of its relational semantics to be expressed. ... In abstract algebra, an interior algebra is an algebraic structure of the signature <A, ·, +, , 0, 1, I> where <A, ·, +, , 0, 1> is a Boolean algebra and I is a unary operator, the interior operator, satisfying the identities: xI ≤ x xII = xI (xy)I = xIyI 1I = 1 xI is called the... Interpretability logics comprise a family of modal logics that extend provability logic to describe interpretability and/or various related metamathematical properties and relations such as weak interpretability, Π1-conservativity, cointerpretability, tolerance, cotolerance, arithmetic complexities. ... Provability logic, or the logic of provability, is a modal logic where the necessity operator is interpreted as provability in a reasonably rich formal theory such as Peano arithmetic. ... Kripke semantics (also known as possible world semantics, relational semantics, or frame semantics) is a formal semantics for modal logic systems, created in late 1950s and early 1960s by Saul Kripke. ...

External links

• 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)

Acknowledgements

This article contains some material originally from the Free On-line Dictionary of Computing which is used with permission under the GFDL. The Free On-line Dictionary of Computing (FOLDOC) is an on-line, searchable encyclopedic dictionary of computing subjects. ...